Matrix calculus - Wikipedia

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class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derivatives_with_vectors"> <div class="vector-toc-text"> 3 Derivatives with vectors </div> </a> <button aria-controls="toc-Derivatives_with_vectors-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Derivatives with vectors subsection </button> <ul id="toc-Derivatives_with_vectors-sublist" class="vector-toc-list"> <li id="toc-Vector-by-scalar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector-by-scalar"> <div class="vector-toc-text"> 3.1 Vector-by-scalar </div> </a> <ul id="toc-Vector-by-scalar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scalar-by-vector" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scalar-by-vector"> <div class="vector-toc-text"> 3.2 Scalar-by-vector </div> </a> <ul id="toc-Scalar-by-vector-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector-by-vector" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector-by-vector"> <div class="vector-toc-text"> 3.3 Vector-by-vector </div> </a> <ul id="toc-Vector-by-vector-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Derivatives_with_matrices" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derivatives_with_matrices"> <div class="vector-toc-text"> 4 Derivatives with matrices </div> </a> <button aria-controls="toc-Derivatives_with_matrices-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Derivatives with matrices subsection </button> <ul id="toc-Derivatives_with_matrices-sublist" class="vector-toc-list"> <li id="toc-Matrix-by-scalar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix-by-scalar"> <div class="vector-toc-text"> 4.1 Matrix-by-scalar </div> </a> <ul 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Layout conventions subsection </button> <ul id="toc-Layout_conventions-sublist" class="vector-toc-list"> <li id="toc-Numerator-layout_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numerator-layout_notation"> <div class="vector-toc-text"> 5.1 Numerator-layout notation </div> </a> <ul id="toc-Numerator-layout_notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Denominator-layout_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Denominator-layout_notation"> <div class="vector-toc-text"> 5.2 Denominator-layout notation </div> </a> <ul id="toc-Denominator-layout_notation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Identities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Identities"> <div class="vector-toc-text"> 6 Identities </div> </a> <button aria-controls="toc-Identities-sublist" class="cdx-button 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searchaux" style="display:none">Specialized notation for multivariable calculus</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Geometric_calculus" title="Geometric calculus">geometric calculus</a> or <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output 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href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d063dc86a53a2efb1fe86f4a5d47d498652766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limits</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuity</a></li></ul> </div><div class="hlist"> <ul><li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul> </div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a> (<a href="/wiki/Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a> <ul><li><a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li> <li><a href="/wiki/Differential_of_a_function" title="Differential of a function">of a function</a></li> <li><a href="/wiki/Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> Concepts</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit differentiation</a></li> <li><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li> <li><a href="/wiki/Inverse_function_rule" title="Inverse function rule">Inverse</a></li> <li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li> <li><a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li> <li><a href="/wiki/Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Integral" title="Integral">Integral</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> (<a href="/wiki/Improper_integral" title="Improper integral">improper</a>)</li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></li> <li><a href="/wiki/Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Integration by</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Integration_by_parts" title="Integration by parts">Parts</a></li> <li><a href="/wiki/Disc_integration" title="Disc integration">Discs</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Cylindrical shells</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a>)</li> <li><a href="/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="/wiki/Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li> <li><a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li> <li><a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li> <li><a href="/wiki/Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Series_(mathematics)" title="Series (mathematics)">Series</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a> (<a href="/wiki/Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li> <a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet</a></li> <li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Theorems</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient_theorem" title="Gradient theorem">Gradient</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">generalized Stokes</a></li> <li><a href="/wiki/Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Formalisms</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a class="mw-selflink selflink">Matrix</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li> <li><a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a></li> <li><a href="/wiki/Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Specialized</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Fractional_calculus" title="Fractional calculus">Fractional</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li> <li><a href="/wiki/Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Miscellanea</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">History</a></li> <li><a href="/wiki/Glossary_of_calculus" title="Glossary of calculus">Glossary</a></li> <li><a href="/wiki/List_of_calculus_topics" title="List of calculus topics">List of topics</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Calculus" title="Template:Calculus"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus" title="Template talk:Calculus"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus" title="Special:EditPage/Template:Calculus"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, matrix calculus is a specialized notation for doing <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a>, especially over spaces of <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>. It collects the various <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> of a single <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> with respect to many <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>, and/or of a <a href="/wiki/Multivariate_function" class="mw-redirect" title="Multivariate function">multivariate function</a> with respect to a single variable, into <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. The notation used here is commonly used in <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>, while the <a href="/wiki/Tensor_index_notation" class="mw-redirect" title="Tensor index notation">tensor index notation</a> is preferred in <a href="/wiki/Physics" title="Physics">physics</a>. Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> with respect to a vector as a <a href="/wiki/Row_and_column_vectors" title="Row and column vectors">column vector or a row vector</a>. Both of these conventions are possible even when the common assumption is made that vectors should be treated as column vectors when combined with matrices (rather than row vectors). A single convention can be somewhat standard throughout a single field that commonly uses matrix calculus (e.g. <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>, statistics, <a href="/wiki/Estimation_theory" title="Estimation theory">estimation theory</a> and <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>). However, even within a given field different authors can be found using competing conventions. Authors of both groups often write as though their specific conventions were standard. Serious mistakes can result when combining results from different authors without carefully verifying that compatible notations have been used. Definitions of these two conventions and comparisons between them are collected in the <a href="#Layout_conventions">layout conventions</a> section. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Scope">Scope</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=1" title="Edit section: Scope">edit</a>]</div> Matrix calculus refers to a number of different notations that use matrices and vectors to collect the derivative of each component of the dependent variable with respect to each component of the independent variable. In general, the independent variable can be a scalar, a vector, or a matrix while the dependent variable can be any of these as well. Each different situation will lead to a different set of rules, or a separate <a href="/wiki/Calculus" title="Calculus">calculus</a>, using the broader sense of the term. Matrix notation serves as a convenient way to collect the many derivatives in an organized way. As a first example, consider the <a href="/wiki/Gradient" title="Gradient">gradient</a> from <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a>. For a scalar function of three independent variables, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{1},x_{2},x_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1},x_{2},x_{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7940f5f72123cd1db44888f39bcb6c8453a21d90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.308ex; height:2.843ex;" alt="{\displaystyle f(x_{1},x_{2},x_{3})}">, the gradient is given by the vector equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f={\frac {\partial f}{\partial x_{1}}}{\hat {x}}_{1}+{\frac {\partial f}{\partial x_{2}}}{\hat {x}}_{2}+{\frac {\partial f}{\partial x_{3}}}{\hat {x}}_{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f={\frac {\partial f}{\partial x_{1}}}{\hat {x}}_{1}+{\frac {\partial f}{\partial x_{2}}}{\hat {x}}_{2}+{\frac {\partial f}{\partial x_{3}}}{\hat {x}}_{3},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8991aace57c93d316ddc9ce50b98eaef5b9d53d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.407ex; height:6.009ex;" alt="{\displaystyle \nabla f={\frac {\partial f}{\partial x_{1}}}{\hat {x}}_{1}+{\frac {\partial f}{\partial x_{2}}}{\hat {x}}_{2}+{\frac {\partial f}{\partial x_{3}}}{\hat {x}}_{3},}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d530f9a088a3c0ee1a3c2dba9eb6e2cba97c790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.509ex;" alt="{\displaystyle {\hat {x}}_{i}}"> represents a unit vector in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"> direction for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i\leq 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6af0d760198175ee2e201862d83f79ce657399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.324ex; height:2.343ex;" alt="{\displaystyle 1\leq i\leq 3}">. This type of generalized derivative can be seen as the derivative of a scalar, f, with respect to a vector, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }">, and its result can be easily collected in vector form. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}={\begin{bmatrix}{\dfrac {\partial f}{\partial x_{1}}}&{\dfrac {\partial f}{\partial x_{2}}}&{\dfrac {\partial f}{\partial x_{3}}}\\\end{bmatrix}}^{\textsf {T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}={\begin{bmatrix}{\dfrac {\partial f}{\partial x_{1}}}&{\dfrac {\partial f}{\partial x_{2}}}&{\dfrac {\partial f}{\partial x_{3}}}\\\end{bmatrix}}^{\textsf {T}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51e875b81a013638f08479249ae9782508cec78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.214ex; height:6.676ex;" alt="{\displaystyle \nabla f=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}={\begin{bmatrix}{\dfrac {\partial f}{\partial x_{1}}}&{\dfrac {\partial f}{\partial x_{2}}}&{\dfrac {\partial f}{\partial x_{3}}}\\\end{bmatrix}}^{\textsf {T}}.}"></dd></dl> More complicated examples include the derivative of a scalar function with respect to a matrix, known as the <a href="#Derivatives_with_matrices">gradient matrix</a>, which collects the derivative with respect to each matrix element in the corresponding position in the resulting matrix. In that case the scalar must be a function of each of the independent variables in the matrix. As another example, if we have an n-vector of dependent variables, or functions, of m independent variables we might consider the derivative of the dependent vector with respect to the independent vector. The result could be collected in an m×n matrix consisting of all of the possible derivative combinations. There are a total of nine possibilities using scalars, vectors, and matrices. Notice that as we consider higher numbers of components in each of the independent and dependent variables we can be left with a very large number of possibilities. The six kinds of derivatives that can be most neatly organized in matrix form are collected in the following table.<a href="#cite_note-minka-1">[1]</a> <table class="wikitable" style="text-align:center; width:35%;"> <caption>Types of matrix derivative </caption> <tbody><tr> <th scope="col">Types </th> <th scope="col">Scalar </th> <th scope="col">Vector </th> <th scope="col">Matrix </th></tr> <tr> <th scope="row">Scalar </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0deac2b96aa5d0329450647f183f9365584c67b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.484ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial x}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d5f2cf89374e95eb31cdf816533244b4d45d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565884c84274a792e9b5af680a30f550eaf5e3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> </td></tr> <tr> <th scope="row">Vector </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a7fae63303065a57b24c2bb67ab80468a24263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734fea892fc38deec1d53fa88abed4ca213c0d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> </td> <td data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na"> </td></tr> <tr> <th scope="row">Matrix </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877eb58a8159dedbc4bc47afc9749803d75d5e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> </td> <td data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na"> </td> <td data-sort-value="" style="background: var(--background-color-interactive, #ececec); color: var(--color-base, inherit); vertical-align: middle; text-align: center;" class="table-na"> </td></tr></tbody></table> Here, we have used the term "matrix" in its most general sense, recognizing that vectors are simply matrices with one column (and scalars are simply vectors with one row). Moreover, we have used bold letters to indicate vectors and bold capital letters for matrices. This notation is used throughout. Notice that we could also talk about the derivative of a vector with respect to a matrix, or any of the other unfilled cells in our table. However, these derivatives are most naturally organized in a <a href="/wiki/Tensor" title="Tensor">tensor</a> of rank higher than 2, so that they do not fit neatly into a matrix. In the following three sections we will define each one of these derivatives and relate them to other branches of mathematics. See the <a href="#Layout_conventions">layout conventions</a> section for a more detailed table. <div class="mw-heading mw-heading3"><h3 id="Relation_to_other_derivatives">Relation to other derivatives</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=2" title="Edit section: Relation to other derivatives">edit</a>]</div> The matrix derivative is a convenient notation for keeping track of partial derivatives for doing calculations. The <a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a> is the standard way in the setting of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> to take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet differentiable, the two derivatives will agree up to translation of notations. As is the case in general for <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a>, some formulae may extend under weaker analytic conditions than the existence of the derivative as approximating linear mapping. <div class="mw-heading mw-heading3"><h3 id="Usages">Usages</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=3" title="Edit section: Usages">edit</a>]</div> Matrix calculus is used for deriving optimal stochastic estimators, often involving the use of <a href="/wiki/Lagrange_multipliers" class="mw-redirect" title="Lagrange multipliers">Lagrange multipliers</a>. This includes the derivation of: <ul><li><a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a></li> <li><a href="/wiki/Wiener_filter" title="Wiener filter">Wiener filter</a></li> <li><a href="/wiki/Expectation-maximization_algorithm#Gaussian_mixture" class="mw-redirect" title="Expectation-maximization algorithm">Expectation-maximization algorithm for Gaussian mixture</a></li> <li><a href="/wiki/Gradient_descent" title="Gradient descent">Gradient descent</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=4" title="Edit section: Notation">edit</a>]</div> The vector and matrix derivatives presented in the sections to follow take full advantage of <a href="/wiki/Matrix_notation" class="mw-redirect" title="Matrix notation">matrix notation</a>, using a single variable to represent a large number of variables. In what follows we will distinguish scalars, vectors and matrices by their typeface. We will let M(n,m) denote the space of <a href="/wiki/Real_number" title="Real number">real</a> n×m matrices with n rows and m columns. Such matrices will be denoted using bold capital letters: A, X, Y, etc. An element of M(n,1), that is, a <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a>, is denoted with a boldface lowercase letter: a, x, y, etc. An element of M(1,1) is a scalar, denoted with lowercase italic typeface: a, t, x, etc. XT denotes matrix <a href="/wiki/Transpose" title="Transpose">transpose</a>, tr(X) is the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a>, and det(X) or |X| is the <a href="/wiki/Determinant" title="Determinant">determinant</a>. All functions are assumed to be of <a href="/wiki/Differentiability_class" class="mw-redirect" title="Differentiability class">differentiability class</a> C1 unless otherwise noted. Generally letters from the first half of the alphabet (a, b, c, ...) will be used to denote constants, and from the second half (t, x, y, ...) to denote variables. NOTE: As mentioned above, there are competing notations for laying out systems of <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> in vectors and matrices, and no standard appears to be emerging yet. The next two introductory sections use the <a href="#Layout_conventions">numerator layout convention</a> simply for the purposes of convenience, to avoid overly complicating the discussion. The section after them discusses <a href="#Layout_conventions">layout conventions</a> in more detail. It is important to realize the following: <ol><li>Despite the use of the terms "numerator layout" and "denominator layout", there are actually more than two possible notational choices involved. The reason is that the choice of numerator vs. denominator (or in some situations, numerator vs. mixed) can be made independently for scalar-by-vector, vector-by-scalar, vector-by-vector, and scalar-by-matrix derivatives, and a number of authors mix and match their layout choices in various ways.</li> <li>The choice of numerator layout in the introductory sections below does not imply that this is the "correct" or "superior" choice. There are advantages and disadvantages to the various layout types. Serious mistakes can result from carelessly combining formulas written in different layouts, and converting from one layout to another requires care to avoid errors. As a result, when working with existing formulas the best policy is probably to identify whichever layout is used and maintain consistency with it, rather than attempting to use the same layout in all situations.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Alternatives">Alternatives</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=5" title="Edit section: Alternatives">edit</a>]</div> The <a href="/wiki/Tensor_index_notation" class="mw-redirect" title="Tensor index notation">tensor index notation</a> with its <a href="/wiki/Einstein_summation" class="mw-redirect" title="Einstein summation">Einstein summation</a> convention is very similar to the matrix calculus, except one writes only a single component at a time. It has the advantage that one can easily manipulate arbitrarily high rank tensors, whereas tensors of rank higher than two are quite unwieldy with matrix notation. All of the work here can be done in this notation without use of the single-variable matrix notation. However, many problems in estimation theory and other areas of applied mathematics would result in too many indices to properly keep track of, pointing in favor of matrix calculus in those areas. Also, Einstein notation can be very useful in proving the identities presented here (see section on <a href="/wiki/Ricci_calculus#Differentiation" title="Ricci calculus">differentiation</a>) as an alternative to typical element notation, which can become cumbersome when the explicit sums are carried around. Note that a matrix can be considered a tensor of rank two. <div class="mw-heading mw-heading2"><h2 id="Derivatives_with_vectors">Derivatives with vectors</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=6" title="Edit section: Derivatives with vectors">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></div> Because vectors are matrices with only one column, the simplest matrix derivatives are vector derivatives. The notations developed here can accommodate the usual operations of <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a> by identifying the space M(n,1) of n-vectors with the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> Rn, and the scalar M(1,1) is identified with R. The corresponding concept from vector calculus is indicated at the end of each subsection. NOTE: The discussion in this section assumes the <a href="#Layout_conventions">numerator layout convention</a> for pedagogical purposes. Some authors use different conventions. The section on <a href="#Layout_conventions">layout conventions</a> discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions. <div class="mw-heading mw-heading3"><h3 id="Vector-by-scalar">Vector-by-scalar</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=7" title="Edit section: Vector-by-scalar">edit</a>]</div> The <a href="/wiki/Derivative" title="Derivative">derivative</a> of a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598107eef2ea088f6bf06ffced12d581ca330e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.798ex; height:3.343ex;" alt="{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}">, by a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> x is written (in <a href="#Layout_conventions">numerator layout notation</a>) as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {y} }{dx}}={\begin{bmatrix}{\frac {dy_{1}}{dx}}\\{\frac {dy_{2}}{dx}}\\\vdots \\{\frac {dy_{m}}{dx}}\\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {y} }{dx}}={\begin{bmatrix}{\frac {dy_{1}}{dx}}\\{\frac {dy_{2}}{dx}}\\\vdots \\{\frac {dy_{m}}{dx}}\\\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf33e0666dc67f681361fe5451120453d7905cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.982ex; margin-bottom: -0.189ex; width:14.897ex; height:17.509ex;" alt="{\displaystyle {\frac {d\mathbf {y} }{dx}}={\begin{bmatrix}{\frac {dy_{1}}{dx}}\\{\frac {dy_{2}}{dx}}\\\vdots \\{\frac {dy_{m}}{dx}}\\\end{bmatrix}}.}"></dd></dl> In <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a> the derivative of a vector y with respect to a scalar x is known as the <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</a> of the vector y, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d5f2cf89374e95eb31cdf816533244b4d45d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}">. Notice here that y: R1 → Rm. Example Simple examples of this include the <a href="/wiki/Velocity" title="Velocity">velocity</a> vector in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, which is the <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</a> of the <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a> vector (considered as a function of time). Also, the <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> is the tangent vector of the velocity. <div class="mw-heading mw-heading3"><h3 id="Scalar-by-vector">Scalar-by-vector</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=8" title="Edit section: Scalar-by-vector">edit</a>]</div> The <a href="/wiki/Derivative" title="Derivative">derivative</a> of a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> y by a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73d2b3df42b37c7d42eab9dd34070aba9feff1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.913ex; height:3.343ex;" alt="{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}">, is written (in <a href="#Layout_conventions">numerator layout notation</a>) as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}={\begin{bmatrix}{\dfrac {\partial y}{\partial x_{1}}}&{\dfrac {\partial y}{\partial x_{2}}}&\cdots &{\dfrac {\partial y}{\partial x_{n}}}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}={\begin{bmatrix}{\dfrac {\partial y}{\partial x_{1}}}&{\dfrac {\partial y}{\partial x_{2}}}&\cdots &{\dfrac {\partial y}{\partial x_{n}}}\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c3aa10868629a9b31eab7d62207549a353c062f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.987ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}={\begin{bmatrix}{\dfrac {\partial y}{\partial x_{1}}}&{\dfrac {\partial y}{\partial x_{2}}}&\cdots &{\dfrac {\partial y}{\partial x_{n}}}\end{bmatrix}}.}"></dd></dl> In <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a>, the <a href="/wiki/Gradient" title="Gradient">gradient</a> of a scalar field f : Rn → R (whose independent coordinates are the components of x) is the transpose of the derivative of a scalar by a vector. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}\\\vdots \\{\frac {\partial f}{\partial x_{n}}}\end{bmatrix}}=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}\\\vdots \\{\frac {\partial f}{\partial x_{n}}}\end{bmatrix}}=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e437ae2c0e37ffa638f57911c2f8b5626152789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:25.275ex; height:13.509ex;" alt="{\displaystyle \nabla f={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}\\\vdots \\{\frac {\partial f}{\partial x_{n}}}\end{bmatrix}}=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\mathsf {T}}}"></dd></dl> By example, in physics, the <a href="/wiki/Electric_field" title="Electric field">electric field</a> is the negative vector <a href="/wiki/Gradient" title="Gradient">gradient</a> of the <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a>. The <a href="/wiki/Directional_derivative" title="Directional derivative">directional derivative</a> of a scalar function f(x) of the space vector x in the direction of the unit vector u (represented in this case as a column vector) is defined using the gradient as follows. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\mathbf {u} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\mathbf {u} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {u} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6667384bf7a845519df9fb5b31d3823699bb4420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.415ex; height:2.843ex;" alt="{\displaystyle \nabla _{\mathbf {u} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {u} }"></dd></dl> Using the notation just defined for the derivative of a scalar with respect to a vector we can re-write the directional derivative as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\mathbf {u} }f={\frac {\partial f}{\partial \mathbf {x} }}\mathbf {u} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\mathbf {u} }f={\frac {\partial f}{\partial \mathbf {x} }}\mathbf {u} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9003675629cc74cbde7790cf5d7f8a80360d3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.293ex; height:5.676ex;" alt="{\displaystyle \nabla _{\mathbf {u} }f={\frac {\partial f}{\partial \mathbf {x} }}\mathbf {u} .}"> This type of notation will be nice when proving product rules and chain rules that come out looking similar to what we are familiar with for the scalar <a href="/wiki/Derivative" title="Derivative">derivative</a>. <div class="mw-heading mw-heading3"><h3 id="Vector-by-vector">Vector-by-vector</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=9" title="Edit section: Vector-by-vector">edit</a>]</div> Each of the previous two cases can be considered as an application of the derivative of a vector with respect to a vector, using a vector of size one appropriately. Similarly we will find that the derivatives involving matrices will reduce to derivatives involving vectors in a corresponding way. The derivative of a <a href="/wiki/Vector_function" class="mw-redirect" title="Vector function">vector function</a> (a vector whose components are functions) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598107eef2ea088f6bf06ffced12d581ca330e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.798ex; height:3.343ex;" alt="{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}">, with respect to an input vector, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73d2b3df42b37c7d42eab9dd34070aba9feff1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.913ex; height:3.343ex;" alt="{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}">, is written (in <a href="#Layout_conventions">numerator layout notation</a>) as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b8c31f61f8b2bbe6f0df4d62012d8ba86ba420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.799ex; margin-bottom: -0.206ex; width:32.837ex; height:19.176ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.}"></dd></dl> In <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a>, the derivative of a vector function y with respect to a vector x whose components represent a space is known as the <a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">pushforward (or differential)</a>, or the <a href="/wiki/Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a>. The pushforward along a vector function f with respect to vector v in Rn is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {f} (\mathbf {v} )={\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}d\mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </mfrac> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {f} (\mathbf {v} )={\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}d\mathbf {v} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e3a027fb57659c1ea87f55b893f330ee456fce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.427ex; height:5.509ex;" alt="{\displaystyle d\mathbf {f} (\mathbf {v} )={\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}d\mathbf {v} .}"> <div class="mw-heading mw-heading2"><h2 id="Derivatives_with_matrices">Derivatives with matrices</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=10" title="Edit section: Derivatives with matrices">edit</a>]</div> There are two types of derivatives with matrices that can be organized into a matrix of the same size. These are the derivative of a matrix by a scalar and the derivative of a scalar by a matrix. These can be useful in minimization problems found in many areas of applied mathematics and have adopted the names tangent matrix and gradient matrix respectively after their analogs for vectors. Note: The discussion in this section assumes the <a href="#Layout_conventions">numerator layout convention</a> for pedagogical purposes. Some authors use different conventions. The section on <a href="#Layout_conventions">layout conventions</a> discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions. <div class="mw-heading mw-heading3"><h3 id="Matrix-by-scalar">Matrix-by-scalar</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=11" title="Edit section: Matrix-by-scalar">edit</a>]</div> The derivative of a matrix function Y by a scalar x is known as the tangent matrix and is given (in <a href="#Layout_conventions">numerator layout notation</a>) by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/968417a1863243432c63633a4636c4bd94008643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.383ex; margin-bottom: -0.288ex; width:35.581ex; height:18.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Scalar-by-matrix">Scalar-by-matrix</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=12" title="Edit section: Scalar-by-matrix">edit</a>]</div> The derivative of a scalar function y, with respect to a p×q matrix X of independent variables, is given (in <a href="#Layout_conventions">numerator layout notation</a>) by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4596c2ed35b0cee2ee78dc35e723728a169bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:34.343ex; height:19.843ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.}"></dd></dl> Important examples of scalar functions of matrices include the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of a matrix and the <a href="/wiki/Determinant" title="Determinant">determinant</a>. In analog with <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a> this derivative is often written as the following. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\mathbf {X} }y(\mathbf {X} )={\frac {\partial y(\mathbf {X} )}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msub> <mi>y</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\mathbf {X} }y(\mathbf {X} )={\frac {\partial y(\mathbf {X} )}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ef85902f8df540af401ecf795336e8310ad0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.817ex; height:5.843ex;" alt="{\displaystyle \nabla _{\mathbf {X} }y(\mathbf {X} )={\frac {\partial y(\mathbf {X} )}{\partial \mathbf {X} }}}"></dd></dl> Also in analog with <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a>, the directional derivative of a scalar f(X) of a matrix X in the direction of matrix Y is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\mathbf {Y} }f=\operatorname {tr} \left({\frac {\partial f}{\partial \mathbf {X} }}\mathbf {Y} \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\mathbf {Y} }f=\operatorname {tr} \left({\frac {\partial f}{\partial \mathbf {X} }}\mathbf {Y} \right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d11114a3031991797488dab8f169e607f1316976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.051ex; height:6.176ex;" alt="{\displaystyle \nabla _{\mathbf {Y} }f=\operatorname {tr} \left({\frac {\partial f}{\partial \mathbf {X} }}\mathbf {Y} \right).}"></dd></dl> It is the gradient matrix, in particular, that finds many uses in minimization problems in <a href="/wiki/Estimation_theory" title="Estimation theory">estimation theory</a>, particularly in the <a href="/wiki/Kalman_filter#Derivations" title="Kalman filter">derivation</a> of the <a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a> algorithm, which is of great importance in the field. <div class="mw-heading mw-heading3"><h3 id="Other_matrix_derivatives">Other matrix derivatives</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=13" title="Edit section: Other matrix derivatives">edit</a>]</div> The three types of derivatives that have not been considered are those involving vectors-by-matrices, matrices-by-vectors, and matrices-by-matrices. These are not as widely considered and a notation is not widely agreed upon. <div class="mw-heading mw-heading2"><h2 id="Layout_conventions">Layout conventions</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=14" title="Edit section: Layout conventions">edit</a>]</div> This section discusses the similarities and differences between notational conventions that are used in the various fields that take advantage of matrix calculus. Although there are largely two consistent conventions, some authors find it convenient to mix the two conventions in forms that are discussed below. After this section, equations will be listed in both competing forms separately. The fundamental issue is that the derivative of a vector with respect to a vector, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734fea892fc38deec1d53fa88abed4ca213c0d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}">, is often written in two competing ways. If the numerator y is of size m and the denominator x of size n, then the result can be laid out as either an m×n matrix or n×m matrix, i.e. the m elements of y laid out in rows and the n elements of x laid out in columns, or vice versa. This leads to the following possibilities: <ol><li>Numerator layout, i.e. lay out according to y and xT (i.e. contrarily to x). This is sometimes known as the Jacobian formulation. This corresponds to the m×n layout in the previous example, which means that the row number of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734fea892fc38deec1d53fa88abed4ca213c0d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> equals to the size of the numerator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb25a040b592282dc2a254c3117e792c3c81161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }"> and the column number of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734fea892fc38deec1d53fa88abed4ca213c0d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> equals to the size of xT.</li> <li>Denominator layout, i.e. lay out according to yT and x (i.e. contrarily to y). This is sometimes known as the Hessian formulation. Some authors term this layout the gradient, in distinction to the Jacobian (numerator layout), which is its transpose. (However, <a href="/wiki/Gradient" title="Gradient">gradient</a> more commonly means the derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aeca299a481abef75e9f37b82688bd388b9beec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.212ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},}"> regardless of layout.). This corresponds to the n×m layout in the previous example, which means that the row number of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734fea892fc38deec1d53fa88abed4ca213c0d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> equals to the size of x (the denominator).</li> <li>A third possibility sometimes seen is to insist on writing the derivative as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} '}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} '}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b8ab287c1ecadb653bf2f278fd48d2a030a63c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.897ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} '}},}"> (i.e. the derivative is taken with respect to the transpose of x) and follow the numerator layout. This makes it possible to claim that the matrix is laid out according to both numerator and denominator. In practice this produces results the same as the numerator layout.</li></ol> When handling the <a href="/wiki/Gradient" title="Gradient">gradient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a7fae63303065a57b24c2bb67ab80468a24263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> and the opposite case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe52817d2eca32570b436933e6fe26e8bc6952a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.212ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},}"> we have the same issues. To be consistent, we should do one of the following: <ol><li>If we choose numerator layout for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9340f03f0542cdc475061049316a4beaa4b492ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.212ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},}"> we should lay out the <a href="/wiki/Gradient" title="Gradient">gradient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a7fae63303065a57b24c2bb67ab80468a24263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> as a row vector, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d5f2cf89374e95eb31cdf816533244b4d45d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> as a column vector.</li> <li>If we choose denominator layout for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9340f03f0542cdc475061049316a4beaa4b492ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.212ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},}"> we should lay out the <a href="/wiki/Gradient" title="Gradient">gradient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a7fae63303065a57b24c2bb67ab80468a24263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> as a column vector, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d5f2cf89374e95eb31cdf816533244b4d45d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> as a row vector.</li> <li>In the third possibility above, we write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} '}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} '}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d38625c96757748baaeb1c9f625c22ea4b78a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.25ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} '}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe52817d2eca32570b436933e6fe26e8bc6952a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.212ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}},}"> and use numerator layout.</li></ol> Not all math textbooks and papers are consistent in this respect throughout. That is, sometimes different conventions are used in different contexts within the same book or paper. For example, some choose denominator layout for gradients (laying them out as column vectors), but numerator layout for the vector-by-vector derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f52635ab92429d4abda0d325d20d8105cb04db6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.212ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}.}"> Similarly, when it comes to scalar-by-matrix derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877eb58a8159dedbc4bc47afc9749803d75d5e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> and matrix-by-scalar derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85bdafd7ed4cebe52fff61221b6adff89228aa8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.821ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}"> then consistent numerator layout lays out according to Y and XT, while consistent denominator layout lays out according to YT and X. In practice, however, following a denominator layout for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85bdafd7ed4cebe52fff61221b6adff89228aa8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.821ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}"> and laying the result out according to YT, is rarely seen because it makes for ugly formulas that do not correspond to the scalar formulas. As a result, the following layouts can often be found: <ol><li>Consistent numerator layout, which lays out <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565884c84274a792e9b5af680a30f550eaf5e3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> according to Y and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877eb58a8159dedbc4bc47afc9749803d75d5e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> according to XT.</li> <li>Mixed layout, which lays out <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565884c84274a792e9b5af680a30f550eaf5e3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> according to Y and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877eb58a8159dedbc4bc47afc9749803d75d5e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> according to X.</li> <li>Use the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} '}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {X} '}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/776fd5369a18e3fe2e8f430cfe6db334cbccd7a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:5.505ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} '}},}"> with results the same as consistent numerator layout.</li></ol> In the following formulas, we handle the five possible combinations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},{\frac {\partial \mathbf {y} }{\partial x}},{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},{\frac {\partial y}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},{\frac {\partial \mathbf {y} }{\partial x}},{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},{\frac {\partial y}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8cf92622bf94d31dbf9e0f249931f28e6646bf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.971ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }},{\frac {\partial \mathbf {y} }{\partial x}},{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }},{\frac {\partial y}{\partial \mathbf {X} }}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565884c84274a792e9b5af680a30f550eaf5e3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> separately. We also handle cases of scalar-by-scalar derivatives that involve an intermediate vector or matrix. (This can arise, for example, if a multi-dimensional <a href="/wiki/Parametric_curve" class="mw-redirect" title="Parametric curve">parametric curve</a> is defined in terms of a scalar variable, and then a derivative of a scalar function of the curve is taken with respect to the scalar that parameterizes the curve.) For each of the various combinations, we give numerator-layout and denominator-layout results, except in the cases above where denominator layout rarely occurs. In cases involving matrices where it makes sense, we give numerator-layout and mixed-layout results. As noted above, cases where vector and matrix denominators are written in transpose notation are equivalent to numerator layout with the denominators written without the transpose. Keep in mind that various authors use different combinations of numerator and denominator layouts for different types of derivatives, and there is no guarantee that an author will consistently use either numerator or denominator layout for all types. Match up the formulas below with those quoted in the source to determine the layout used for that particular type of derivative, but be careful not to assume that derivatives of other types necessarily follow the same kind of layout. When taking derivatives with an aggregate (vector or matrix) denominator in order to find a maximum or minimum of the aggregate, it should be kept in mind that using numerator layout will produce results that are transposed with respect to the aggregate. For example, in attempting to find the <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> estimate of a <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a> using matrix calculus, if the domain is a k×1 column vector, then the result using the numerator layout will be in the form of a 1×k row vector. Thus, either the results should be transposed at the end or the denominator layout (or mixed layout) should be used. <dl><dd><table class="wikitable"> <caption>Result of differentiating various kinds of aggregates with other kinds of aggregates </caption> <tbody><tr> <th colspan="2" rowspan="2"> </th> <th colspan="2">Scalar y </th> <th colspan="2">Column vector y (size m×1) </th> <th colspan="2">Matrix Y (size m×n) </th></tr> <tr> <th>Notation</th> <th>Type </th> <th>Notation</th> <th>Type </th> <th>Notation</th> <th>Type </th></tr> <tr> <th rowspan="2">Scalar x </th> <th>Numerator </th> <td rowspan="2" style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0deac2b96aa5d0329450647f183f9365584c67b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.484ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial x}}}"> </td> <td rowspan="2">Scalar </td> <td rowspan="2" style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d5f2cf89374e95eb31cdf816533244b4d45d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> </td> <td>Size-m <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a> </td> <td rowspan="2" style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565884c84274a792e9b5af680a30f550eaf5e3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> </td> <td>m×n matrix </td></tr> <tr> <th>Denominator </th> <td>Size-m <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vector</a> </td> <td> </td></tr> <tr> <th rowspan="2">Column vector x (size n×1) </th> <th>Numerator </th> <td rowspan="2" style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a7fae63303065a57b24c2bb67ab80468a24263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}}"> </td> <td>Size-n <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vector</a> </td> <td rowspan="2" style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734fea892fc38deec1d53fa88abed4ca213c0d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> </td> <td>m×n matrix </td> <td rowspan="2" style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/433f2d2da465f5a3f2aa8dff5c9d6dd8e9947eef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial \mathbf {x} }}}"> </td> <td rowspan="2"> </td></tr> <tr> <th>Denominator </th> <td>Size-n <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a> </td> <td>n×m matrix </td></tr> <tr> <th rowspan="2">Matrix X (size p×q) </th> <th>Numerator </th> <td rowspan="2" style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877eb58a8159dedbc4bc47afc9749803d75d5e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> </td> <td>q×p matrix </td> <td rowspan="2" style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a7d5bedcc1bc202bd55040b26137a6c1740850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {X} }}}"> </td> <td rowspan="2"> </td> <td rowspan="2" style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7d1744e8920b3885bde9168c70643df3a49cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial \mathbf {X} }}}"> </td> <td rowspan="2"> </td></tr> <tr> <th>Denominator </th> <td>p×q matrix </td></tr></tbody></table></dd></dl> The results of operations will be transposed when switching between numerator-layout and denominator-layout notation. <div class="mw-heading mw-heading3"><h3 id="Numerator-layout_notation">Numerator-layout notation</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=15" title="Edit section: Numerator-layout notation">edit</a>]</div> Using numerator-layout notation, we have:<a href="#cite_note-minka-1">[1]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}&{\frac {\partial y}{\partial x_{2}}}&\cdots &{\frac {\partial y}{\partial x_{n}}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}\\{\frac {\partial y_{2}}{\partial x}}\\\vdots \\{\frac {\partial y_{m}}{\partial x}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}&{\frac {\partial y}{\partial x_{2}}}&\cdots &{\frac {\partial y}{\partial x_{n}}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}\\{\frac {\partial y_{2}}{\partial x}}\\\vdots \\{\frac {\partial y_{m}}{\partial x}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4dab37ae19b5f0cf0d0eb6572f79414823309e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -31.005ex; width:35.095ex; height:63.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}&{\frac {\partial y}{\partial x_{2}}}&\cdots &{\frac {\partial y}{\partial x_{n}}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}\\{\frac {\partial y_{2}}{\partial x}}\\\vdots \\{\frac {\partial y_{m}}{\partial x}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}}"></dd></dl> The following definitions are only provided in numerator-layout notation: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\partial \mathbf {Y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.\\d\mathbf {X} &={\begin{bmatrix}dx_{11}&dx_{12}&\cdots &dx_{1n}\\dx_{21}&dx_{22}&\cdots &dx_{2n}\\\vdots &\vdots &\ddots &\vdots \\dx_{m1}&dx_{m2}&\cdots &dx_{mn}\\\end{bmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial \mathbf {Y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.\\d\mathbf {X} &={\begin{bmatrix}dx_{11}&dx_{12}&\cdots &dx_{1n}\\dx_{21}&dx_{22}&\cdots &dx_{2n}\\\vdots &\vdots &\ddots &\vdots \\dx_{m1}&dx_{m2}&\cdots &dx_{mn}\\\end{bmatrix}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24abba682ae88d29a1c0d2be6c7908004827bd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.838ex; width:37.761ex; height:32.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial \mathbf {Y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}.\\d\mathbf {X} &={\begin{bmatrix}dx_{11}&dx_{12}&\cdots &dx_{1n}\\dx_{21}&dx_{22}&\cdots &dx_{2n}\\\vdots &\vdots &\ddots &\vdots \\dx_{m1}&dx_{m2}&\cdots &dx_{mn}\\\end{bmatrix}}.\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Denominator-layout_notation">Denominator-layout notation</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=16" title="Edit section: Denominator-layout notation">edit</a>]</div> Using denominator-layout notation, we have:<a href="#cite_note-2">[2]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}\\{\frac {\partial y}{\partial x_{2}}}\\\vdots \\{\frac {\partial y}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}&{\frac {\partial y_{2}}{\partial x}}&\cdots &{\frac {\partial y_{m}}{\partial x}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{1}}}\\{\frac {\partial y_{1}}{\partial x_{2}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{1}}{\partial x_{n}}}&{\frac {\partial y_{2}}{\partial x_{n}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{12}}}&\cdots &{\frac {\partial y}{\partial x_{1q}}}\\{\frac {\partial y}{\partial x_{21}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{2q}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{p1}}}&{\frac {\partial y}{\partial x_{p2}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>q</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}\\{\frac {\partial y}{\partial x_{2}}}\\\vdots \\{\frac {\partial y}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}&{\frac {\partial y_{2}}{\partial x}}&\cdots &{\frac {\partial y_{m}}{\partial x}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{1}}}\\{\frac {\partial y_{1}}{\partial x_{2}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{1}}{\partial x_{n}}}&{\frac {\partial y_{2}}{\partial x_{n}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{12}}}&\cdots &{\frac {\partial y}{\partial x_{1q}}}\\{\frac {\partial y}{\partial x_{21}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{2q}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{p1}}}&{\frac {\partial y}{\partial x_{p2}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3a1156c8e1eb6940b3d89e84436602a5d3b3b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -31.018ex; margin-bottom: -0.32ex; width:35.05ex; height:63.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial y}{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}\\{\frac {\partial y}{\partial x_{2}}}\\\vdots \\{\frac {\partial y}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial x}}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}&{\frac {\partial y_{2}}{\partial x}}&\cdots &{\frac {\partial y_{m}}{\partial x}}\end{bmatrix}}.\\{\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}&={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{1}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{1}}}\\{\frac {\partial y_{1}}{\partial x_{2}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{1}}{\partial x_{n}}}&{\frac {\partial y_{2}}{\partial x_{n}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}.\\{\frac {\partial y}{\partial \mathbf {X} }}&={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{12}}}&\cdots &{\frac {\partial y}{\partial x_{1q}}}\\{\frac {\partial y}{\partial x_{21}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{2q}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{p1}}}&{\frac {\partial y}{\partial x_{p2}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}.\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Identities">Identities</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=17" title="Edit section: Identities">edit</a>]</div> As noted above, in general, the results of operations will be transposed when switching between numerator-layout and denominator-layout notation. To help make sense of all the identities below, keep in mind the most important rules: the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a>, <a href="/wiki/Product_rule" title="Product rule">product rule</a> and <a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">sum rule</a>. The sum rule applies universally, and the product rule applies in most of the cases below, provided that the order of matrix products is maintained, since matrix products are not commutative. The chain rule applies in some of the cases, but unfortunately does not apply in matrix-by-scalar derivatives or scalar-by-matrix derivatives (in the latter case, mostly involving the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> operator applied to matrices). In the latter case, the product rule can't quite be applied directly, either, but the equivalent can be done with a bit more work using the differential identities. The following identities adopt the following conventions: <ul><li>the scalars, a, b, c, d, and e are constant in respect of, and the scalars, u, and v are functions of one of x, x, or X;</li> <li>the vectors, a, b, c, d, and e are constant in respect of, and the vectors, u, and v are functions of one of x, x, or X;</li> <li>the matrices, A, B, C, D, and E are constant in respect of, and the matrices, U and V are functions of one of x, x, or X.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Vector-by-vector_identities">Vector-by-vector identities</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=18" title="Edit section: Vector-by-vector identities">edit</a>]</div> This is presented first because all of the operations that apply to vector-by-vector differentiation apply directly to vector-by-scalar or scalar-by-vector differentiation simply by reducing the appropriate vector in the numerator or denominator to a scalar. <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Identities: vector-by-vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/734fea892fc38deec1d53fa88abed4ca213c0d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}}"> </caption> <tbody><tr> <th scope="col" width="150">Condition </th> <th scope="col" width="10">Expression </th> <th scope="col" width="100">Numerator layout, i.e. by y and xT </th> <th scope="col" width="100">Denominator layout, i.e. by yT and x </th></tr> <tr> <td>a is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {a} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {a} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/958b2d5f89c535a52ce7fff9e05fe4e627a2d968" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.018ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {a} }{\partial \mathbf {x} }}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"> </td></tr> <tr> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {x} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {x} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d662096885d61e6ad2cb2b072266eeb92b4bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.018ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {x} }{\partial \mathbf {x} }}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a458c8aeb096ce732abf346ae8edf3e4f53a126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }"> </td></tr> <tr> <td>A is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78536a28b60c910d163274b0b34d45fa031e9bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.038ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad17fef126eb0106f72750a43341adc6a6d3f45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} ^{\top }}"> </td></tr> <tr> <td>A is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d833cdc6909b4d0d04c15cd7ad8603ff27ae8e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.549ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} }{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad17fef126eb0106f72750a43341adc6a6d3f45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} ^{\top }}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"> </td></tr> <tr> <td>a is not a function of x, u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial a\mathbf {u} }{\partial \,\mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial a\mathbf {u} }{\partial \,\mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dfecbf40fe626ab3127ce8a746b6634a994ad5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.323ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial a\mathbf {u} }{\partial \,\mathbf {x} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc3848a0f754779903a5fcbfca880e547131822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.869ex; height:5.509ex;" alt="{\displaystyle a{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> </td></tr> <tr> <td>v = v(x), a is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial v\mathbf {a} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial v\mathbf {a} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f70133239a41b9eee1fadd3478310a35dd0dcd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.035ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial v\mathbf {a} }{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} {\frac {\partial v}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} {\frac {\partial v}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d306ccdbb4217ee301c29811ce2868314c2f8cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.865ex; height:5.509ex;" alt="{\displaystyle \mathbf {a} {\frac {\partial v}{\partial \mathbf {x} }}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial v}{\partial \mathbf {x} }}\mathbf {a} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial v}{\partial \mathbf {x} }}\mathbf {a} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e721d0c95ec288899ac3193456e53a49ece198" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.375ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial v}{\partial \mathbf {x} }}\mathbf {a} ^{\top }}"> </td></tr> <tr> <td>v = v(x), u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial v\mathbf {u} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial v\mathbf {u} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2066a38a9ca34fa949b245a30f9db21d0a1ad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.22ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial v\mathbf {u} }{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+\mathbf {u} {\frac {\partial v}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+\mathbf {u} {\frac {\partial v}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/892704def36f1cc64f11a6c0215ec51ef9eab1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.658ex; height:5.509ex;" alt="{\displaystyle v{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+\mathbf {u} {\frac {\partial v}{\partial \mathbf {x} }}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial v}{\partial \mathbf {x} }}\mathbf {u} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial v}{\partial \mathbf {x} }}\mathbf {u} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d78e00037f6acca83a6adb1e20c97bcf554cf8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.169ex; height:5.509ex;" alt="{\displaystyle v{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial v}{\partial \mathbf {x} }}\mathbf {u} ^{\top }}"> </td></tr> <tr> <td>A is not a function of x, u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {A} \mathbf {u} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {A} \mathbf {u} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90176ceeba25834221588db998405fbf496f28a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.112ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {A} \mathbf {u} }{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca59fa17c64462705923f9d19a94855145d17ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.659ex; height:5.509ex;" alt="{\displaystyle \mathbf {A} {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb82a947253010a263ed4129686a13624cd3913" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.17ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} ^{\top }}"> </td></tr> <tr> <td>u = u(x), v = v(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a04b4ac30ef0436c3370c5edab3f0d4a199c7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.153ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial \mathbf {x} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52f38f3ef23184ecff65c5021bfbb871fc90862" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.045ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}+{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> </td></tr> <tr> <td>u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c109547638fa59cebf9f7108bb45e25f3195c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.239ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1e4e09be5b951e7211f5e5c5cec24db94aa650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.425ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5760ce585e22ca85b3e2eb0bed73ec93972d7c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.425ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}"> </td></tr> <tr> <td>u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} (\mathbf {u} ))}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} (\mathbf {u} ))}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4a4732d8310a92a9da1df520b0881c6c2a2391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.101ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} (\mathbf {u} ))}{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863f4469144d2838cebf0cc17220580b587de814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.778ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c13f5c3adf9648b50529a6f614e222ed547f1169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.778ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}"> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Scalar-by-vector_identities">Scalar-by-vector identities</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=19" title="Edit section: Scalar-by-vector identities">edit</a>]</div> The fundamental identities are placed above the thick black line. <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Identities: scalar-by-vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}=\nabla _{\mathbf {x} }y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}=\nabla _{\mathbf {x} }y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90980ce250ba28c3c15c56d4d11266174d6d5de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.985ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}=\nabla _{\mathbf {x} }y}"> </caption> <tbody><tr> <th scope="col" width="150">Condition </th> <th scope="col" width="200">Expression </th> <th scope="col" width="200">Numerator layout, i.e. by xT; result is row vector </th> <th scope="col" width="200">Denominator layout, i.e. by x; result is column vector </th></tr> <tr> <td>a is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial a}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>a</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial a}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32812f2bb30add57b4eee8d2395c22171e4cd0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.018ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial a}{\partial \mathbf {x} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e911cc971717ee3efb356101c08eb7d639f1e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.847ex; height:2.676ex;" alt="{\displaystyle \mathbf {0} ^{\top }}"> <a href="#cite_note-zerovec-3">[nb 1]</a></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"> <a href="#cite_note-zerovec-3">[nb 1]</a> </td></tr> <tr> <td>a is not a function of x, u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial au}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>a</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial au}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6afe994d94b9afaeb2f5be107b3cc80c7d14e597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.167ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial au}{\partial \mathbf {x} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {\partial u}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {\partial u}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a810aaae2f563dff3076167676aad45912d6141e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.795ex; height:5.509ex;" alt="{\displaystyle a{\frac {\partial u}{\partial \mathbf {x} }}}"> </td></tr> <tr> <td>u = u(x), v = v(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (u+v)}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (u+v)}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d090c428a46cdf4c689c6aab67132613b12c23d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.714ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (u+v)}{\partial \mathbf {x} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial \mathbf {x} }}+{\frac {\partial v}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial \mathbf {x} }}+{\frac {\partial v}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/173128cc2201bfbe5dd4d212aa4c9b85a8901c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.971ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial u}{\partial \mathbf {x} }}+{\frac {\partial v}{\partial \mathbf {x} }}}"> </td></tr> <tr> <td>u = u(x), v = v(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial uv}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial uv}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62bbe649c4c10b40ad7c285f683c14d0d19f1586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.065ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial uv}{\partial \mathbf {x} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u{\frac {\partial v}{\partial \mathbf {x} }}+v{\frac {\partial u}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u{\frac {\partial v}{\partial \mathbf {x} }}+v{\frac {\partial u}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61377155e3e4e645bb7da7407e3471c977f3374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.428ex; height:5.509ex;" alt="{\displaystyle u{\frac {\partial v}{\partial \mathbf {x} }}+v{\frac {\partial u}{\partial \mathbf {x} }}}"> </td></tr> <tr> <td>u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial g(u)}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g(u)}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/529210ef3bb05c491ca23d16fd569de19aaf2790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.862ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial g(u)}{\partial \mathbf {x} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbdd0b3a5604012f3e36b0a043e14eda0473b40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.974ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}}"> </td></tr> <tr> <td>u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f(g(u))}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f(g(u))}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af45dbaf32ddbac178b025a7104514f68854031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.95ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial f(g(u))}{\partial \mathbf {x} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f47b671c6863eec0c30e3ca2c13c8b44cda940f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.332ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {x} }}}"> </td></tr> <tr> <td>u = u(x), v = v(x) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\top }\mathbf {v} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\top }\mathbf {v} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2825d4660a2877088eae13ac9212014c94686ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.652ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\top }\mathbf {v} }{\partial \mathbf {x} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ^{\top }{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ^{\top }{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0909255166321ba8c3a3c905fb3eadbb6f7d8353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.963ex; height:5.509ex;" alt="{\displaystyle \mathbf {u} ^{\top }{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea808c56c3bef8e0062eb9a3e770faff8ca7e810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.239ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> in numerator layout </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {u} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21582a81bf63af93cedfe219a0cafabafa3f873e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.941ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {u} }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea808c56c3bef8e0062eb9a3e770faff8ca7e810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.239ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> in denominator layout </td></tr> <tr> <td>u = u(x), v = v(x), A is not a function of x </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {A} \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\top }\mathbf {A} \mathbf {v} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {A} \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\top }\mathbf {A} \mathbf {v} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f918d23f5e2faf5d6d63350add0c0b44069d9420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.691ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {A} \mathbf {v} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {u} ^{\top }\mathbf {A} \mathbf {v} }{\partial \mathbf {x} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ^{\top }\mathbf {A} {\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\top }\mathbf {A} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ^{\top }\mathbf {A} {\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\top }\mathbf {A} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/492c14c8694462d93cf8fda9b955023fd379fc9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.513ex; height:5.509ex;" alt="{\displaystyle \mathbf {u} ^{\top }\mathbf {A} {\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}+\mathbf {v} ^{\top }\mathbf {A} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea808c56c3bef8e0062eb9a3e770faff8ca7e810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.239ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> in numerator layout </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} \mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {A} ^{\top }\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} \mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {A} ^{\top }\mathbf {u} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfae76269eac1993ebee42d6c666eb97521667ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.491ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {A} \mathbf {v} +{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}\mathbf {A} ^{\top }\mathbf {u} }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea808c56c3bef8e0062eb9a3e770faff8ca7e810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.239ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }},{\frac {\partial \mathbf {v} }{\partial \mathbf {x} }}}"> in denominator layout </td></tr> <tr> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}f}{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}f}{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c114c179bd4a220ce5f9116b38e7b967aeea854" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.258ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}f}{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa5c20429694e37b0e2a8e88236718c3a7846aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.602ex; height:2.676ex;" alt="{\displaystyle \mathbf {H} ^{\top }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f017b876ed763037d8818ec5dfbbdc6703e0f683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }">, the <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a><a href="#cite_note-matrix-cookbook-4">[3]</a> </td></tr> <tr style="border-top: 3px solid;"> <td>a is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {a} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial (\mathbf {x} \cdot \mathbf {a} )}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {a} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial (\mathbf {x} \cdot \mathbf {a} )}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6ed15892178203127464eb3e49d96428fdba09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.258ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {a} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial (\mathbf {x} \cdot \mathbf {a} )}{\partial \mathbf {x} }}=}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {x} }{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\top }\mathbf {a} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {x} }{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\top }\mathbf {a} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9468f11961e533d7eac44cf59f8a67c303838d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.303ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {x} }{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\top }\mathbf {a} }{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e658506721d522ab65157ebcd09bf3680e0280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.81ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} ^{\top }}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.299ex; height:1.676ex;" alt="{\displaystyle \mathbf {a} }"> </td></tr> <tr> <td>A is not a function of x b is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {b} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {b} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33746d9c237151235e998787ce7483d6ea174f26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.034ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {b} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} ^{\top }\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} ^{\top }\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5a9188c9936aa6fa0201177514404d68b75c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.016ex; height:2.676ex;" alt="{\displaystyle \mathbf {b} ^{\top }\mathbf {A} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ^{\top }\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ^{\top }\mathbf {b} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e28fe8970315522056b99313d26f01e21378809c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.016ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} ^{\top }\mathbf {b} }"> </td></tr> <tr> <td>A is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f79fc2ad2d79eb056e19b953579e639bd908c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.96ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfa8333aecea1751ef23947834554d5120ea8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.829ex; height:3.343ex;" alt="{\displaystyle \mathbf {x} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {A} +\mathbf {A} ^{\top }\right)\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {A} +\mathbf {A} ^{\top }\right)\mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4276a822f4dd02c9cb3a81fa3f864d2e5fc89a3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.318ex; height:3.343ex;" alt="{\displaystyle \left(\mathbf {A} +\mathbf {A} ^{\top }\right)\mathbf {x} }"> </td></tr> <tr> <td>A is not a function of x A is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f79fc2ad2d79eb056e19b953579e639bd908c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.96ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbf {x} ^{\top }\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbf {x} ^{\top }\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f04a82dae42a166db9f0a8fc00a560fca56f5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.104ex; height:2.676ex;" alt="{\displaystyle 2\mathbf {x} ^{\top }\mathbf {A} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbf {A} \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbf {A} \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7d8369411e2fbead8deaa98e903bdfb50e4c8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.593ex; height:2.176ex;" alt="{\displaystyle 2\mathbf {A} \mathbf {x} }"> </td></tr> <tr> <td>A is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea64b23cf32c0f1b517db44e11c0ad79a5106419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.039ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}\mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} +\mathbf {A} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} +\mathbf {A} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f64aa97fdc75a6d1dcb908a58fffd9491f00f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.39ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} +\mathbf {A} ^{\top }}"> </td></tr> <tr> <td>A is not a function of x A is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea64b23cf32c0f1b517db44e11c0ad79a5106419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.039ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}\mathbf {x} ^{\top }\mathbf {A} \mathbf {x} }{\partial \mathbf {x} \partial \mathbf {x} ^{\top }}}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2edd8e22f4248bca5bcbda34e01d5d98c84c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.182ex; height:2.176ex;" alt="{\displaystyle 2\mathbf {A} }"> </td></tr> <tr> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {x} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\top }\mathbf {x} }{\partial \mathbf {x} }}={\frac {\partial \left\Vert \mathbf {x} \right\Vert ^{2}}{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {x} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\top }\mathbf {x} }{\partial \mathbf {x} }}={\frac {\partial \left\Vert \mathbf {x} \right\Vert ^{2}}{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e1cf268e085df1ee83aac96dfff55306963712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.546ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial (\mathbf {x} \cdot \mathbf {x} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {x} ^{\top }\mathbf {x} }{\partial \mathbf {x} }}={\frac {\partial \left\Vert \mathbf {x} \right\Vert ^{2}}{\partial \mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbf {x} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbf {x} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e2a20f833b8109efb7f2aaf4e94cd9396434c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.084ex; height:2.676ex;" alt="{\displaystyle 2\mathbf {x} ^{\top }}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc93e64e2fd853638df47579a82b776492538a94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.573ex; height:2.176ex;" alt="{\displaystyle 2\mathbf {x} }"> </td></tr> <tr> <td>a is not a function of x, u = u(x) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {a} \cdot \mathbf {u} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {a} ^{\top }\mathbf {u} }{\partial \mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {a} \cdot \mathbf {u} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {a} ^{\top }\mathbf {u} }{\partial \mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a29f01a14e83256d03e9c29285db0d940a34b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.429ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {a} \cdot \mathbf {u} )}{\partial \mathbf {x} }}={\frac {\partial \mathbf {a} ^{\top }\mathbf {u} }{\partial \mathbf {x} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4d78ab25dff8ed0afe7236d4acfe6b91408024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.45ex; height:5.509ex;" alt="{\displaystyle \mathbf {a} ^{\top }{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/019cfb07566d5eaa08d51e7ade58319013287d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.639ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> in numerator layout </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {a} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f776aa21151b780272ca2d96d6b80f2918787e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.939ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {a} }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/019cfb07566d5eaa08d51e7ade58319013287d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.639ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}}"> in denominator layout </td></tr> <tr> <td>a, b are not functions of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \;{\textbf {a}}^{\top }{\textbf {x}}{\textbf {x}}^{\top }{\textbf {b}}}{\partial \;{\textbf {x}}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mspace width="thickmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">a</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \;{\textbf {a}}^{\top }{\textbf {x}}{\textbf {x}}^{\top }{\textbf {b}}}{\partial \;{\textbf {x}}}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1edd2b075e08e9a351aef9a9a35f7b97745b6a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.881ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \;{\textbf {a}}^{\top }{\textbf {x}}{\textbf {x}}^{\top }{\textbf {b}}}{\partial \;{\textbf {x}}}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {x}}^{\top }\left({\textbf {a}}{\textbf {b}}^{\top }+{\textbf {b}}{\textbf {a}}^{\top }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">a</mtext> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">a</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {x}}^{\top }\left({\textbf {a}}{\textbf {b}}^{\top }+{\textbf {b}}{\textbf {a}}^{\top }\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7ff694699b3ff52c797f0cd22095ea312b288b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.87ex; height:3.343ex;" alt="{\displaystyle {\textbf {x}}^{\top }\left({\textbf {a}}{\textbf {b}}^{\top }+{\textbf {b}}{\textbf {a}}^{\top }\right)}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\textbf {a}}{\textbf {b}}^{\top }+{\textbf {b}}{\textbf {a}}^{\top }\right){\textbf {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">a</mtext> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">a</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\textbf {a}}{\textbf {b}}^{\top }+{\textbf {b}}{\textbf {a}}^{\top }\right){\textbf {x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0061a0b9fa51ef5acf057a38e4bebd1994428a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.359ex; height:3.343ex;" alt="{\displaystyle \left({\textbf {a}}{\textbf {b}}^{\top }+{\textbf {b}}{\textbf {a}}^{\top }\right){\textbf {x}}}"> </td></tr> <tr> <td>A, b, C, D, e are not functions of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \;({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\top }{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})}{\partial \;{\textbf {x}}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">A</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">C</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">D</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">e</mtext> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \;({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\top }{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})}{\partial \;{\textbf {x}}}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64332881101c98ff26b4a5ddca8c58f1706bfbaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.596ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial \;({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\top }{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})}{\partial \;{\textbf {x}}}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\textbf {D}}{\textbf {x}}+{\textbf {e}})^{\top }{\textbf {C}}^{\top }{\textbf {A}}+({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\top }{\textbf {C}}{\textbf {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">D</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">e</mtext> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">C</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">A</mtext> </mrow> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">A</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">C</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">D</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\textbf {D}}{\textbf {x}}+{\textbf {e}})^{\top }{\textbf {C}}^{\top }{\textbf {A}}+({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\top }{\textbf {C}}{\textbf {D}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca615e10c05a1e568864cdc6aa3b4a7bf3920201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.205ex; height:3.176ex;" alt="{\displaystyle ({\textbf {D}}{\textbf {x}}+{\textbf {e}})^{\top }{\textbf {C}}^{\top }{\textbf {A}}+({\textbf {A}}{\textbf {x}}+{\textbf {b}})^{\top }{\textbf {C}}{\textbf {D}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {D}}^{\top }{\textbf {C}}^{\top }({\textbf {A}}{\textbf {x}}+{\textbf {b}})+{\textbf {A}}^{\top }{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">D</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">C</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">A</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">b</mtext> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">A</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">C</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">D</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">e</mtext> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {D}}^{\top }{\textbf {C}}^{\top }({\textbf {A}}{\textbf {x}}+{\textbf {b}})+{\textbf {A}}^{\top }{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c15cfc9944d406776ae445c00b7edb3a24960e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.205ex; height:3.176ex;" alt="{\displaystyle {\textbf {D}}^{\top }{\textbf {C}}^{\top }({\textbf {A}}{\textbf {x}}+{\textbf {b}})+{\textbf {A}}^{\top }{\textbf {C}}({\textbf {D}}{\textbf {x}}+{\textbf {e}})}"> </td></tr> <tr> <td>a is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \;\|\mathbf {x} -\mathbf {a} \|}{\partial \;\mathbf {x} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \;\|\mathbf {x} -\mathbf {a} \|}{\partial \;\mathbf {x} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f79f54e0fdc756b54b7159d473e88c92442c5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.128ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \;\|\mathbf {x} -\mathbf {a} \|}{\partial \;\mathbf {x} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(\mathbf {x} -\mathbf {a} )^{\top }}{\|\mathbf {x} -\mathbf {a} \|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(\mathbf {x} -\mathbf {a} )^{\top }}{\|\mathbf {x} -\mathbf {a} \|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1781a14d271703a21c02db543f6f15c7504689e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.707ex; height:6.676ex;" alt="{\displaystyle {\frac {(\mathbf {x} -\mathbf {a} )^{\top }}{\|\mathbf {x} -\mathbf {a} \|}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathbf {x} -\mathbf {a} }{\|\mathbf {x} -\mathbf {a} \|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathbf {x} -\mathbf {a} }{\|\mathbf {x} -\mathbf {a} \|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc11625ac9a7a8a601cb34914aedd3fa136da0ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.712ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathbf {x} -\mathbf {a} }{\|\mathbf {x} -\mathbf {a} \|}}}"> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Vector-by-scalar_identities">Vector-by-scalar identities</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=20" title="Edit section: Vector-by-scalar identities">edit</a>]</div> <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Identities: vector-by-scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d5f2cf89374e95eb31cdf816533244b4d45d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.565ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}"> </caption> <tbody><tr> <th scope="col" width="150">Condition </th> <th scope="col" width="100">Expression </th> <th scope="col" width="100">Numerator layout, i.e. by y, result is column vector </th> <th scope="col" width="100">Denominator layout, i.e. by yT, result is row vector </th></tr> <tr> <td>a is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {a} }{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {a} }{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2403c3cc72bd3ed553acb923bc7447f6d385186d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.937ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {a} }{\partial x}}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"><a href="#cite_note-zerovec-3">[nb 1]</a> </td></tr> <tr> <td>a is not a function of x, u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial a\mathbf {u} }{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial a\mathbf {u} }{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e80ff9b8e89ea05d8db43f69ce34de371c1cf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.323ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial a\mathbf {u} }{\partial x}}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {\partial \mathbf {u} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {\partial \mathbf {u} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b433304fd60b85cd17c6ed3e67c3e8bc4ccecb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.869ex; height:5.509ex;" alt="{\displaystyle a{\frac {\partial \mathbf {u} }{\partial x}}}"> </td></tr> <tr> <td>A is not a function of x, u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {A} \mathbf {u} }{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {A} \mathbf {u} }{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82a51057a9c87af4b56541f0ceb7b72c5503a76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.112ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {A} \mathbf {u} }{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} {\frac {\partial \mathbf {u} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} {\frac {\partial \mathbf {u} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5da2652a2f0bed384134aee5a93da723c492cdd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.659ex; height:5.509ex;" alt="{\displaystyle \mathbf {A} {\frac {\partial \mathbf {u} }{\partial x}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}\mathbf {A} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}\mathbf {A} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0001aaf5c61e6f28843cb06d7e00c33f6497572e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.17ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}\mathbf {A} ^{\top }}"> </td></tr> <tr> <td>u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} ^{\top }}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} ^{\top }}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a85dd1bc5550254a747d886a96600a271fb6a121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.604ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} ^{\top }}{\partial x}}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial \mathbf {u} }{\partial x}}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \mathbf {u} }{\partial x}}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/670d8a3d03cfbae768f35cb15e09ab33cbc3128a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.571ex; height:6.676ex;" alt="{\displaystyle \left({\frac {\partial \mathbf {u} }{\partial x}}\right)^{\top }}"> </td></tr> <tr> <td>u = u(x), v = v(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/328c66df01aedcdcdc0e657acd7c4c051e6c244c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.153ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {u} +\mathbf {v} )}{\partial x}}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}+{\frac {\partial \mathbf {v} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}+{\frac {\partial \mathbf {v} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05340877809cb74e26dd0cad81c1145575fd64c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.045ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}+{\frac {\partial \mathbf {v} }{\partial x}}}"> </td></tr> <tr> <td>u = u(x), v = v(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {u} ^{\top }\times \mathbf {v} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {u} ^{\top }\times \mathbf {v} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36d2d891ffacc41fafe0db135906825394bcdd17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.664ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial (\mathbf {u} ^{\top }\times \mathbf {v} )}{\partial x}}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial \mathbf {u} }{\partial x}}\right)^{\top }\times \mathbf {v} +\mathbf {u} ^{\top }\times {\frac {\partial \mathbf {v} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \mathbf {u} }{\partial x}}\right)^{\top }\times \mathbf {v} +\mathbf {u} ^{\top }\times {\frac {\partial \mathbf {v} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae20debc8fb7f32ce65f7187969d2aca7a5926a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.065ex; height:6.676ex;" alt="{\displaystyle \left({\frac {\partial \mathbf {u} }{\partial x}}\right)^{\top }\times \mathbf {v} +\mathbf {u} ^{\top }\times {\frac {\partial \mathbf {v} }{\partial x}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}\times \mathbf {v} +\mathbf {u} ^{\top }\times \left({\frac {\partial \mathbf {v} }{\partial x}}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>×</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}\times \mathbf {v} +\mathbf {u} ^{\top }\times \left({\frac {\partial \mathbf {v} }{\partial x}}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5a6ac92bc47f834f7a15a04bcf5c610d85c30a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.065ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}\times \mathbf {v} +\mathbf {u} ^{\top }\times \left({\frac {\partial \mathbf {v} }{\partial x}}\right)^{\top }}"> </td></tr> <tr> <td rowspan="2">u = u(x)</td> <td rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22cfa430afd2b53011ac4481e16663612975ab13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.239ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e8ee2418e721db2af180b3d76d1efa568f2fdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.425ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9138e5afab8b0b56e70ae6c4c8c474585851024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.425ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}"> </td></tr> <tr> <td colspan="2">Assumes consistent matrix layout; see below. </td></tr> <tr> <td rowspan="2">u = u(x)</td> <td rowspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} (\mathbf {u} ))}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} (\mathbf {u} ))}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c723775f576375055a8c4856aabfd02d5442ef67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.101ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} (\mathbf {u} ))}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/647ef44fa309881b717e72ac62c3d3e792ca1f77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.778ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {u} }{\partial x}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbda51fdc9e577ea838eff94b37a7d5bd5cec7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.778ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial x}}{\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}{\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}"> </td></tr> <tr> <td colspan="2">Assumes consistent matrix layout; see below. </td></tr> <tr> <td>U = U(x), v = v(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {U} \times \mathbf {v} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {U} \times \mathbf {v} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f73d01da7c15f77f13266bf31b8fd6441723c055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.725ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {U} \times \mathbf {v} )}{\partial x}}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {U} }{\partial x}}\times \mathbf {v} +\mathbf {U} \times {\frac {\partial \mathbf {v} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {U} }{\partial x}}\times \mathbf {v} +\mathbf {U} \times {\frac {\partial \mathbf {v} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38af13f061f9dade2facf921a7015020beccf065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.765ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {U} }{\partial x}}\times \mathbf {v} +\mathbf {U} \times {\frac {\partial \mathbf {v} }{\partial x}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ^{\top }\times \left({\frac {\partial \mathbf {U} }{\partial x}}\right)+{\frac {\partial \mathbf {v} }{\partial x}}\times \mathbf {U} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>×</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ^{\top }\times \left({\frac {\partial \mathbf {U} }{\partial x}}\right)+{\frac {\partial \mathbf {v} }{\partial x}}\times \mathbf {U} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b31bca78bd80c5167676cd3f1e0b103e8d90c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.208ex; height:6.176ex;" alt="{\displaystyle \mathbf {v} ^{\top }\times \left({\frac {\partial \mathbf {U} }{\partial x}}\right)+{\frac {\partial \mathbf {v} }{\partial x}}\times \mathbf {U} ^{\top }}"> </td></tr></tbody></table></dd></dl> NOTE: The formulas involving the vector-by-vector derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6b3a3b83b8ac0194801afdbafcb2dbb049ceaab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.785ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {g} (\mathbf {u} )}{\partial \mathbf {u} }}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5e50a2602de3257a1f112ed7abe5fed3c7c51b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.353ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial \mathbf {f} (\mathbf {g} )}{\partial \mathbf {g} }}}"> (whose outputs are matrices) assume the matrices are laid out consistent with the vector layout, i.e. numerator-layout matrix when numerator-layout vector and vice versa; otherwise, transpose the vector-by-vector derivatives. <div class="mw-heading mw-heading3"><h3 id="Scalar-by-matrix_identities">Scalar-by-matrix identities</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=21" title="Edit section: Scalar-by-matrix identities">edit</a>]</div> Note that exact equivalents of the scalar <a href="/wiki/Product_rule" title="Product rule">product rule</a> and <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> do not exist when applied to matrix-valued functions of matrices. However, the product rule of this sort does apply to the differential form (see below), and this is the way to derive many of the identities below involving the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> function, combined with the fact that the trace function allows transposing and cyclic permutation, i.e.: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} )&=\operatorname {tr} \left(\mathbf {A^{\top }} \right)\\\operatorname {tr} (\mathbf {ABCD} )&=\operatorname {tr} (\mathbf {BCDA} )=\operatorname {tr} (\mathbf {CDAB} )=\operatorname {tr} (\mathbf {DABC} )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} )&=\operatorname {tr} \left(\mathbf {A^{\top }} \right)\\\operatorname {tr} (\mathbf {ABCD} )&=\operatorname {tr} (\mathbf {BCDA} )=\operatorname {tr} (\mathbf {CDAB} )=\operatorname {tr} (\mathbf {DABC} )\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/601cfc285e5b24a76d75b903a7fe9f2ca2f0330f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:56.155ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} )&=\operatorname {tr} \left(\mathbf {A^{\top }} \right)\\\operatorname {tr} (\mathbf {ABCD} )&=\operatorname {tr} (\mathbf {BCDA} )=\operatorname {tr} (\mathbf {CDAB} )=\operatorname {tr} (\mathbf {DABC} )\end{aligned}}}"></dd></dl> For example, to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AXBX^{\top }C} )}{\partial \mathbf {X} }}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AXBX^{\top }C} )}{\partial \mathbf {X} }}:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e27f1397c079cb5813621259b11913f43ff1d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.86ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AXBX^{\top }C} )}{\partial \mathbf {X} }}:}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}d\operatorname {tr} (\mathbf {AXBX^{\top }C} )&=d\operatorname {tr} \left(\mathbf {CAXBX^{\top }} \right)=\operatorname {tr} \left(d\left(\mathbf {CAXBX^{\top }} \right)\right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAX} d(\mathbf {BX^{\top }} \right)+d\left(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAX} d\left(\mathbf {BX^{\top }} \right)\right)+\operatorname {tr} \left(d(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAXB} d\left(\mathbf {X^{\top }} \right)\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)+\operatorname {tr} (\mathbf {CA} \left(d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)^{\top }\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left((d\mathbf {X} )\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} \right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} (d\mathbf {X} )\right)+\operatorname {tr} \left(\mathbf {BX^{\top }} \mathbf {CA} (d\mathbf {X} )\right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }} \mathbf {CA} \right)d\mathbf {X} \right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} \right)^{\top }d\mathbf {X} \right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.73em 0.73em 0.73em 0.73em 0.73em 0.73em 0.73em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>d</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>d</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mi>d</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> <mi>d</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">X</mi> <msup> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d\operatorname {tr} (\mathbf {AXBX^{\top }C} )&=d\operatorname {tr} \left(\mathbf {CAXBX^{\top }} \right)=\operatorname {tr} \left(d\left(\mathbf {CAXBX^{\top }} \right)\right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAX} d(\mathbf {BX^{\top }} \right)+d\left(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAX} d\left(\mathbf {BX^{\top }} \right)\right)+\operatorname {tr} \left(d(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAXB} d\left(\mathbf {X^{\top }} \right)\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)+\operatorname {tr} (\mathbf {CA} \left(d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)^{\top }\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left((d\mathbf {X} )\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} \right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} (d\mathbf {X} )\right)+\operatorname {tr} \left(\mathbf {BX^{\top }} \mathbf {CA} (d\mathbf {X} )\right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }} \mathbf {CA} \right)d\mathbf {X} \right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} \right)^{\top }d\mathbf {X} \right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca1c0807b52b0c32354803abbd029430fdf8813" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -23.005ex; width:64.598ex; height:47.176ex;" alt="{\displaystyle {\begin{aligned}d\operatorname {tr} (\mathbf {AXBX^{\top }C} )&=d\operatorname {tr} \left(\mathbf {CAXBX^{\top }} \right)=\operatorname {tr} \left(d\left(\mathbf {CAXBX^{\top }} \right)\right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAX} d(\mathbf {BX^{\top }} \right)+d\left(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAX} d\left(\mathbf {BX^{\top }} \right)\right)+\operatorname {tr} \left(d(\mathbf {CAX} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAXB} d\left(\mathbf {X^{\top }} \right)\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)+\operatorname {tr} (\mathbf {CA} \left(d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {CAXB} (d\mathbf {X} )^{\top }\right)^{\top }\right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left((d\mathbf {X} )\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} \right)+\operatorname {tr} \left(\mathbf {CA} (d\mathbf {X} )\mathbf {BX^{\top }} \right)\\[1ex]&=\operatorname {tr} \left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} (d\mathbf {X} )\right)+\operatorname {tr} \left(\mathbf {BX^{\top }} \mathbf {CA} (d\mathbf {X} )\right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }} \mathbf {CA} \right)d\mathbf {X} \right)\\[1ex]&=\operatorname {tr} \left(\left(\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} \right)^{\top }d\mathbf {X} \right)\end{aligned}}}"> Therefore, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }CA} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">C</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }CA} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7286d89dabca44109b71421a306fb7ee559c6bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:47.771ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} +\mathbf {BX^{\top }CA} .}"> (numerator layout)</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">C</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">X</mi> <msup> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d22c1d74be5c5899e4ae05fa7b28b934aa6d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.75ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=\mathbf {CAXB} +\mathbf {A^{\top }C^{\top }XB^{\top }} .}"> (denominator layout)</dd></dl> (For the last step, see the <a href="#convert_differential_derivative">Conversion from differential to derivative form</a> section.) <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Identities: scalar-by-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877eb58a8159dedbc4bc47afc9749803d75d5e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}}"> </caption> <tbody><tr> <th scope="col" width="175">Condition </th> <th scope="col" width="10">Expression </th> <th scope="col" width="100">Numerator layout, i.e. by XT </th> <th scope="col" width="100">Denominator layout, i.e. by X </th></tr> <tr> <td>a is not a function of X</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial a}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>a</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial a}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a4fdd67f3c5a1d0173f0a334b84f8a9a59b2ad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.627ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial a}{\partial \mathbf {X} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e911cc971717ee3efb356101c08eb7d639f1e03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.847ex; height:2.676ex;" alt="{\displaystyle \mathbf {0} ^{\top }}"> <a href="#cite_note-zeromatrix-5">[nb 2]</a></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"> <a href="#cite_note-zeromatrix-5">[nb 2]</a> </td></tr> <tr> <td>a is not a function of X, u = u(X)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial au}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>a</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial au}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ce0b07d088bb02c86f450147600a8bbf5b1eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.167ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial au}{\partial \mathbf {X} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {\partial u}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {\partial u}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fed7d7d28bdbc2744d62d867efd75ded4125491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.404ex; height:5.509ex;" alt="{\displaystyle a{\frac {\partial u}{\partial \mathbf {X} }}}"> </td></tr> <tr> <td>u = u(X), v = v(X)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (u+v)}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (u+v)}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62d3b10ed72e9621108cba7002b9898ec639032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.714ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (u+v)}{\partial \mathbf {X} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial \mathbf {X} }}+{\frac {\partial v}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial \mathbf {X} }}+{\frac {\partial v}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0983a16b735d1e662a7ca8fb0611c2988b3c6b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.188ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial u}{\partial \mathbf {X} }}+{\frac {\partial v}{\partial \mathbf {X} }}}"> </td></tr> <tr> <td>u = u(X), v = v(X)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial uv}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial uv}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ea16a8464c0639e9908fdf1135d557ea9664a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.065ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial uv}{\partial \mathbf {X} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u{\frac {\partial v}{\partial \mathbf {X} }}+v{\frac {\partial u}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u{\frac {\partial v}{\partial \mathbf {X} }}+v{\frac {\partial u}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47c3425f58086c3dbb710f0d37e925ab8e5570ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.645ex; height:5.509ex;" alt="{\displaystyle u{\frac {\partial v}{\partial \mathbf {X} }}+v{\frac {\partial u}{\partial \mathbf {X} }}}"> </td></tr> <tr> <td>u = u(X)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial g(u)}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g(u)}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a9928c51c85394a70af5e52fa78b51d0d4b5ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.862ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial g(u)}{\partial \mathbf {X} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a17151869852ed9ffe7a40e0cce78996085e8421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.583ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}}"> </td></tr> <tr> <td>u = u(X)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f(g(u))}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f(g(u))}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539dcba6b651b08f718995e2e223e28127547f29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.95ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial f(g(u))}{\partial \mathbf {X} }}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739f0ab5eae7008fa8406c37b5ed481400416efe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.941ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial f(g)}{\partial g}}{\frac {\partial g(u)}{\partial u}}{\frac {\partial u}{\partial \mathbf {X} }}}"> </td></tr> <tr> <td rowspan="2">U = U(X)</td> <td rowspan="2"><a href="#cite_note-matrix-cookbook-4">[3]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial g(\mathbf {U} )}{\partial X_{ij}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g(\mathbf {U} )}{\partial X_{ij}}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e1cf77eb519d2b66c4702d800fa969c2439458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.589ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial g(\mathbf {U} )}{\partial X_{ij}}}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f12fc649233d340a17f3f8f180b5a53ad56da33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.929ex; height:6.509ex;" alt="{\displaystyle \operatorname {tr} \left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\top }{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\top }{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2517d9eec03083c6bb0fc9694c13a2fac1e84e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.121ex; height:7.509ex;" alt="{\displaystyle \operatorname {tr} \left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\top }{\frac {\partial \mathbf {U} }{\partial X_{ij}}}\right)}"> </td></tr> <tr> <td colspan="2">Both forms assume numerator layout for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {U} }{\partial X_{ij}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {U} }{\partial X_{ij}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d080d462ad085fc3fdf76fd920b03cc0e33492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:6.202ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \mathbf {U} }{\partial X_{ij}}},}"> i.e. mixed layout if denominator layout for X is being used. </td></tr> <tr style="border-top: 3px solid;"> <td>a and b are not functions of X</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {X} \mathbf {b} }{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {X} \mathbf {b} }{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e0dc9e873739d1cbea6cce77e9f3ce186e5c449" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.923ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {X} \mathbf {b} }{\partial \mathbf {X} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} \mathbf {a} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} \mathbf {a} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/723538c20cdfdf08d367551e9ebad157185c11b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.296ex; height:2.676ex;" alt="{\displaystyle \mathbf {b} \mathbf {a} ^{\top }}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \mathbf {b} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \mathbf {b} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5212e425c514fa403abd4d77b9a35f3fbb56e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.296ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} \mathbf {b} ^{\top }}"> </td></tr> <tr> <td>a and b are not functions of X</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {X} ^{\top }\mathbf {b} }{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {X} ^{\top }\mathbf {b} }{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd3b71509260c850ad7fb403e4bd87c8b7432f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.433ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {a} ^{\top }\mathbf {X} ^{\top }\mathbf {b} }{\partial \mathbf {X} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \mathbf {b} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \mathbf {b} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5212e425c514fa403abd4d77b9a35f3fbb56e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.296ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} \mathbf {b} ^{\top }}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} \mathbf {a} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} \mathbf {a} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/723538c20cdfdf08d367551e9ebad157185c11b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.296ex; height:2.676ex;" alt="{\displaystyle \mathbf {b} \mathbf {a} ^{\top }}"> </td></tr> <tr> <td>a and b are not functions of X, f(v) is a real-valued differentiable function </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f(\mathbf {Xa+b} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">a</mi> <mo mathvariant="bold">+</mo> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f(\mathbf {Xa+b} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1d80086df99345c85b1a898c06014f8135a39a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.609ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial f(\mathbf {Xa+b} )}{\partial \mathbf {X} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} {\frac {\partial f}{\partial \mathbf {v} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} {\frac {\partial f}{\partial \mathbf {v} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12ae34a9d4472532b27cddd70e85d648a0e54c48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.865ex; height:5.676ex;" alt="{\displaystyle \mathbf {a} {\frac {\partial f}{\partial \mathbf {v} }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\mathbf {a} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\mathbf {a} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc5e93737fa38a37aaf26aec444359f0045b23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.375ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\mathbf {a} ^{\top }}"> </td></tr> <tr> <td>a, b and C are not functions of X</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {X} \mathbf {a} +\mathbf {b} )^{\top }\mathbf {C} (\mathbf {X} \mathbf {a} +\mathbf {b} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {X} \mathbf {a} +\mathbf {b} )^{\top }\mathbf {C} (\mathbf {X} \mathbf {a} +\mathbf {b} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f6ed60df94a3b8f976b12bc841d526f70fa847d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.957ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial (\mathbf {X} \mathbf {a} +\mathbf {b} )^{\top }\mathbf {C} (\mathbf {X} \mathbf {a} +\mathbf {b} )}{\partial \mathbf {X} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\left(\mathbf {C} +\mathbf {C} ^{\top }\right)(\mathbf {X} \mathbf {a} +\mathbf {b} )\mathbf {a} ^{\top }\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\left(\mathbf {C} +\mathbf {C} ^{\top }\right)(\mathbf {X} \mathbf {a} +\mathbf {b} )\mathbf {a} ^{\top }\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ccb559d29f7babc5ba60ca42693f27e6830b01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.635ex; height:3.843ex;" alt="{\displaystyle \left(\left(\mathbf {C} +\mathbf {C} ^{\top }\right)(\mathbf {X} \mathbf {a} +\mathbf {b} )\mathbf {a} ^{\top }\right)^{\top }}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {C} +\mathbf {C} ^{\top }\right)(\mathbf {X} \mathbf {a} +\mathbf {b} )\mathbf {a} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {C} +\mathbf {C} ^{\top }\right)(\mathbf {X} \mathbf {a} +\mathbf {b} )\mathbf {a} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482744c9730a4290989c74fce8d810a661844a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.995ex; height:3.343ex;" alt="{\displaystyle \left(\mathbf {C} +\mathbf {C} ^{\top }\right)(\mathbf {X} \mathbf {a} +\mathbf {b} )\mathbf {a} ^{\top }}"> </td></tr> <tr> <td>a, b and C are not functions of X</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {X} \mathbf {a} )^{\top }\mathbf {C} (\mathbf {X} \mathbf {b} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {X} \mathbf {a} )^{\top }\mathbf {C} (\mathbf {X} \mathbf {b} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa10f60baa44c01c957d89966d3c85b0c012cc03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.492ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial (\mathbf {X} \mathbf {a} )^{\top }\mathbf {C} (\mathbf {X} \mathbf {b} )}{\partial \mathbf {X} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {C} \mathbf {X} \mathbf {b} \mathbf {a} ^{\top }+\mathbf {C} ^{\top }\mathbf {X} \mathbf {a} \mathbf {b} ^{\top }\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {C} \mathbf {X} \mathbf {b} \mathbf {a} ^{\top }+\mathbf {C} ^{\top }\mathbf {X} \mathbf {a} \mathbf {b} ^{\top }\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc109bd989b0d1a309d96bace7e7fb2d8e7f5dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.484ex; height:3.843ex;" alt="{\displaystyle \left(\mathbf {C} \mathbf {X} \mathbf {b} \mathbf {a} ^{\top }+\mathbf {C} ^{\top }\mathbf {X} \mathbf {a} \mathbf {b} ^{\top }\right)^{\top }}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} \mathbf {X} \mathbf {b} \mathbf {a} ^{\top }+\mathbf {C} ^{\top }\mathbf {X} \mathbf {a} \mathbf {b} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} \mathbf {X} \mathbf {b} \mathbf {a} ^{\top }+\mathbf {C} ^{\top }\mathbf {X} \mathbf {a} \mathbf {b} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf492f202f61442516eb8e043f604b4c4df230a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.844ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} \mathbf {X} \mathbf {b} \mathbf {a} ^{\top }+\mathbf {C} ^{\top }\mathbf {X} \mathbf {a} \mathbf {b} ^{\top }}"> </td></tr> <tr style="border-top: 3px solid;"> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {X} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {X} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a9805fa9f6d7a52892f27adf35e6251107fc4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.64ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {X} )}{\partial \mathbf {X} }}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a458c8aeb096ce732abf346ae8edf3e4f53a126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }"> </td></tr> <tr> <td>U = U(X), V = V(X)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {U} +\mathbf {V} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {U} +\mathbf {V} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7013455247c8b324328ac929ba4f1e7098ee0d4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.537ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {U} +\mathbf {V} )}{\partial \mathbf {X} }}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {U} )}{\partial \mathbf {X} }}+{\frac {\partial \operatorname {tr} (\mathbf {V} )}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {U} )}{\partial \mathbf {X} }}+{\frac {\partial \operatorname {tr} (\mathbf {V} )}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df7495eaa3dc319c2a71d8f6c6e6f8e0fe0fc4c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.25ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {U} )}{\partial \mathbf {X} }}+{\frac {\partial \operatorname {tr} (\mathbf {V} )}{\partial \mathbf {X} }}}"> </td></tr> <tr> <td>a is not a function of X, U = U(X)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (a\mathbf {U} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (a\mathbf {U} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b69e42228c596eeb6e0ef0dc510eec07d613e9be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.907ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (a\mathbf {U} )}{\partial \mathbf {X} }}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {\partial \operatorname {tr} (\mathbf {U} )}{\partial \mathbf {X} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {\partial \operatorname {tr} (\mathbf {U} )}{\partial \mathbf {X} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1695732d358e009de551c7128f0c9455ac994dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.453ex; height:5.843ex;" alt="{\displaystyle a{\frac {\partial \operatorname {tr} (\mathbf {U} )}{\partial \mathbf {X} }}}"> </td></tr> <tr> <td>g(X) is any <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. eX, sin(X), cos(X), ln(X), etc. using a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>); g(x) is the equivalent scalar function, g<big>′</big>(x) is its derivative, and g<big>′</big>(X) is the corresponding matrix function</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {g(X)} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold">X</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {g(X)} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a064db6435da7854cb867e5ccc3a80d41d1d09e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.055ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {g(X)} )}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {g} '(\mathbf {X} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} '(\mathbf {X} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7f7e46ceffd1ae7b7dd3dd611a5ad0b7bf6c24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.85ex; height:3.009ex;" alt="{\displaystyle \mathbf {g} '(\mathbf {X} )}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {g} '(\mathbf {X} )\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {g} '(\mathbf {X} )\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/004f3f3d488ddf3e727b7606e72aa8e462430b76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.17ex; height:3.509ex;" alt="{\displaystyle \left(\mathbf {g} '(\mathbf {X} )\right)^{\top }}"> </td></tr> <tr> <td>A is not a function of X</td> <td><a href="#cite_note-6">[4]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AX} )}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} (\mathbf {XA} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AX} )}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} (\mathbf {XA} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84d1aa8af8253d42ae369364c54dd78de3fc23d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.963ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AX} )}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} (\mathbf {XA} )}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad17fef126eb0106f72750a43341adc6a6d3f45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} ^{\top }}"> </td></tr> <tr> <td>A is not a function of X</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AX^{\top }} \right)}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} \left(\mathbf {X^{\top }A} \right)}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">A</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AX^{\top }} \right)}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} \left(\mathbf {X^{\top }A} \right)}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8c0fc9ec670de5fe23b4246fe2d76be44f95fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.626ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AX^{\top }} \right)}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} \left(\mathbf {X^{\top }A} \right)}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad17fef126eb0106f72750a43341adc6a6d3f45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.53ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} ^{\top }}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"> </td></tr> <tr> <td>A is not a function of X</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {X^{\top }AX} \right)}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {X^{\top }AX} \right)}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9126b954bbf4c379b9163124115f588b3fb0c6ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.51ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {X^{\top }AX} \right)}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70021d3dcc7fe381170fe3bc19e496d007e1ccc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.437ex; height:3.343ex;" alt="{\displaystyle \mathbf {X} ^{\top }\left(\mathbf {A} +\mathbf {A} ^{\top }\right)}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {A} +\mathbf {A} ^{\top }\right)\mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {A} +\mathbf {A} ^{\top }\right)\mathbf {X} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace084727d5180e3bedd94a8adf52ac3e9926f2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.927ex; height:3.343ex;" alt="{\displaystyle \left(\mathbf {A} +\mathbf {A} ^{\top }\right)\mathbf {X} }"> </td></tr> <tr> <td>A is not a function of X</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {X^{-1}A} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">−</mo> <mn mathvariant="bold">1</mn> </mrow> </msup> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {X^{-1}A} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caeaa0e998f5dd93c8c24da0201356233ade7084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.306ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {X^{-1}A} )}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathbf {X} ^{-1}\mathbf {A} \mathbf {X} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mathbf {X} ^{-1}\mathbf {A} \mathbf {X} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc305e5231e2f7f58a7732c3a14f7afd35e6bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.532ex; height:2.843ex;" alt="{\displaystyle -\mathbf {X} ^{-1}\mathbf {A} \mathbf {X} ^{-1}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\left(\mathbf {X} ^{-1}\right)^{\top }\mathbf {A} ^{\top }\left(\mathbf {X} ^{-1}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\left(\mathbf {X} ^{-1}\right)^{\top }\mathbf {A} ^{\top }\left(\mathbf {X} ^{-1}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6706a280f8f26ee7b215b8b1e62d744c8fa35c60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.324ex; height:3.843ex;" alt="{\displaystyle -\left(\mathbf {X} ^{-1}\right)^{\top }\mathbf {A} ^{\top }\left(\mathbf {X} ^{-1}\right)^{\top }}"> </td></tr> <tr> <td>A, B are not functions of X</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AXB} )}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} (\mathbf {BAX} )}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AXB} )}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} (\mathbf {BAX} )}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3bbf8c5b52412cb79e54943d0bc96302c46f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.766ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {AXB} )}{\partial \mathbf {X} }}={\frac {\partial \operatorname {tr} (\mathbf {BAX} )}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {BA} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {BA} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a18141ac6f5bc712127885259d5e08c9f70cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.921ex; height:2.176ex;" alt="{\displaystyle \mathbf {BA} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A^{\top }B^{\top }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A^{\top }B^{\top }} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f884ef5234ded216ad9c85d01804c2230561bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.942ex; height:2.676ex;" alt="{\displaystyle \mathbf {A^{\top }B^{\top }} }"> </td></tr> <tr> <td>A, B, C are not functions of X</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">C</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d308ef2cf41872ed850c976533ceddc3ae3a8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.342ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {AXBX^{\top }C} \right)}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {BX^{\top }CA} +\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {BX^{\top }CA} +\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bb14e4a55acd3268ccb5c2aec7ed92b01926fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.137ex; height:2.843ex;" alt="{\displaystyle \mathbf {BX^{\top }CA} +\mathbf {B^{\top }X^{\top }A^{\top }C^{\top }} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A^{\top }C^{\top }XB^{\top }} +\mathbf {CAXB} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">X</mi> <msup> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A^{\top }C^{\top }XB^{\top }} +\mathbf {CAXB} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/731db524a29eb7f7cfd96aebf3465342a20951fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.115ex; height:2.843ex;" alt="{\displaystyle \mathbf {A^{\top }C^{\top }XB^{\top }} +\mathbf {CAXB} }"> </td></tr> <tr> <td>n is a positive integer</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {X} ^{n}\right)}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {X} ^{n}\right)}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592a6c60e64cf3213f850ee3d1a7029e896c787b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.858ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {X} ^{n}\right)}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\mathbf {X} ^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\mathbf {X} ^{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10c30dce861a77cd0ecc22693199d885b3636f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.733ex; height:2.676ex;" alt="{\displaystyle n\mathbf {X} ^{n-1}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\left(\mathbf {X} ^{n-1}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\left(\mathbf {X} ^{n-1}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c4c74f1858ddb4e953f1ac03cb120c64829b38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.374ex; height:3.843ex;" alt="{\displaystyle n\left(\mathbf {X} ^{n-1}\right)^{\top }}"> </td></tr> <tr> <td>A is not a function of X, n is a positive integer</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {A} \mathbf {X} ^{n}\right)}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {A} \mathbf {X} ^{n}\right)}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb7ee2c37af38ef0ca41f10596f139937a7ed1cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.878ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} \left(\mathbf {A} \mathbf {X} ^{n}\right)}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n-1}\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n-1}\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/632dca1e7d4f83d90fc35a52963d911aedb6bb86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.765ex; height:7.343ex;" alt="{\displaystyle \sum _{i=0}^{n-1}\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n-1}\left(\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n-1}\left(\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad084f1997b89b9588017dba88492e9038eb09bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.406ex; height:7.343ex;" alt="{\displaystyle \sum _{i=0}^{n-1}\left(\mathbf {X} ^{i}\mathbf {A} \mathbf {X} ^{n-i-1}\right)^{\top }}"> </td></tr> <tr> <td></td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} \left(e^{\mathbf {X} }\right)}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} \left(e^{\mathbf {X} }\right)}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc3401b8c223a95078bc7f1a8869d3da7a81433" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.684ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} \left(e^{\mathbf {X} }\right)}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\mathbf {X} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\mathbf {X} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed793a9b89ae133ff2ca88bd57d8f6a0d6b7c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.744ex; height:2.676ex;" alt="{\displaystyle e^{\mathbf {X} }}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(e^{\mathbf {X} }\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(e^{\mathbf {X} }\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00471b04ad3fb96f3652979a68f338082556c3af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.384ex; height:3.843ex;" alt="{\displaystyle \left(e^{\mathbf {X} }\right)^{\top }}"> </td></tr> <tr> <td></td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\sin(\mathbf {X} ))}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\sin(\mathbf {X} ))}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddea54e39ff138ec9edf5ace47ed67734f1534ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.305ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\sin(\mathbf {X} ))}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\mathbf {X} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\mathbf {X} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbecf7f94cfc852a806951cfea3721357efc13c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.94ex; height:2.843ex;" alt="{\displaystyle \cos(\mathbf {X} )}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cos(\mathbf {X} ))^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cos(\mathbf {X} ))^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13d06784cd0f58568474c8bec8c22d21642a54c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.26ex; height:3.176ex;" alt="{\displaystyle (\cos(\mathbf {X} ))^{\top }}"> </td></tr> <tr style="border-top: 3px solid;"> <td></td> <td><a href="#cite_note-7">[5]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial |\mathbf {X} |}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial |\mathbf {X} |}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f8a3dd4df3a384d18f8c238e040367ed910244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.921ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial |\mathbf {X} |}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cofactor} (X)^{\top }=|\mathbf {X} |\mathbf {X} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cofactor</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cofactor} (X)^{\top }=|\mathbf {X} |\mathbf {X} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/530cf3bae33500eca60cb74dc6e9cc35af7807e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.144ex; height:3.176ex;" alt="{\displaystyle \operatorname {cofactor} (X)^{\top }=|\mathbf {X} |\mathbf {X} ^{-1}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cofactor} (X)=|\mathbf {X} |\left(\mathbf {X} ^{-1}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cofactor</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cofactor} (X)=|\mathbf {X} |\left(\mathbf {X} ^{-1}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6e7da1452a07f2758c8b00212f636ea9c6313e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.274ex; height:3.843ex;" alt="{\displaystyle \operatorname {cofactor} (X)=|\mathbf {X} |\left(\mathbf {X} ^{-1}\right)^{\top }}"> </td></tr> <tr> <td>a is not a function of X</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \ln |a\mathbf {X} |}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \ln |a\mathbf {X} |}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4b71e11a0e1cb42a0713a57e606b702d51c51d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.864ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \ln |a\mathbf {X} |}{\partial \mathbf {X} }}=}"><a href="#cite_note-8">[nb 3]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3a8d8fbfd1c467c634e0b3811a6314cb51067e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.352ex; height:2.676ex;" alt="{\displaystyle \mathbf {X} ^{-1}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {X} ^{-1}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {X} ^{-1}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4481a312241ed7076947dd275d748c1484034c2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.993ex; height:3.843ex;" alt="{\displaystyle \left(\mathbf {X} ^{-1}\right)^{\top }}"> </td></tr> <tr> <td>A, B are not functions of X</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>     <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial |\mathbf {AXB} |}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial |\mathbf {AXB} |}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe8072a1a0026d8d14e6cfffbcbe8cf5b4dfb07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.841ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial |\mathbf {AXB} |}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {AXB} |\mathbf {X} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {AXB} |\mathbf {X} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75ecaa99b491711c537293f4e1957f850a7198e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.586ex; height:3.176ex;" alt="{\displaystyle |\mathbf {AXB} |\mathbf {X} ^{-1}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {AXB} |\left(\mathbf {X} ^{-1}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {AXB} |\left(\mathbf {X} ^{-1}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6b20230d8e87b88d40da338f61eff42482ea9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.227ex; height:3.843ex;" alt="{\displaystyle |\mathbf {AXB} |\left(\mathbf {X} ^{-1}\right)^{\top }}"> </td></tr> <tr> <td>n is a positive integer</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \left|\mathbf {X} ^{n}\right|}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>|</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \left|\mathbf {X} ^{n}\right|}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/310fadc8718d089b500a7ce341470f6a577d4bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.526ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \left|\mathbf {X} ^{n}\right|}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\left|\mathbf {X} ^{n}\right|\mathbf {X} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow> <mo>|</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\left|\mathbf {X} ^{n}\right|\mathbf {X} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8fc23b0d906532ffb65c964b9e867902a907db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.053ex; height:3.176ex;" alt="{\displaystyle n\left|\mathbf {X} ^{n}\right|\mathbf {X} ^{-1}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\left|\mathbf {X} ^{n}\right|\left(\mathbf {X} ^{-1}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow> <mo>|</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>|</mo> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\left|\mathbf {X} ^{n}\right|\left(\mathbf {X} ^{-1}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10dd003a4b091c05eb98d704e112d328eb83b6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.693ex; height:3.843ex;" alt="{\displaystyle n\left|\mathbf {X} ^{n}\right|\left(\mathbf {X} ^{-1}\right)^{\top }}"> </td></tr> <tr> <td>(see <a href="/wiki/Pseudo-inverse" class="mw-redirect" title="Pseudo-inverse">pseudo-inverse</a>)</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>      <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \ln \left|\mathbf {X} ^{\top }\mathbf {X} \right|}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \ln \left|\mathbf {X} ^{\top }\mathbf {X} \right|}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5efa13e5bc3efc9a40fdf502b7385815936ad3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.777ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial \ln \left|\mathbf {X} ^{\top }\mathbf {X} \right|}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbf {X} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbf {X} ^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec58d500f78bb2b3e4b8de14fa239efd373591f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.693ex; height:2.509ex;" alt="{\displaystyle 2\mathbf {X} ^{+}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\left(\mathbf {X} ^{+}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\left(\mathbf {X} ^{+}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba39efe825c5bc9e28a863c630243f68ae1dbd7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.333ex; height:3.676ex;" alt="{\displaystyle 2\left(\mathbf {X} ^{+}\right)^{\top }}"> </td></tr> <tr> <td>(see <a href="/wiki/Pseudo-inverse" class="mw-redirect" title="Pseudo-inverse">pseudo-inverse</a>)</td> <td><a href="#cite_note-matrix-cookbook-4">[3]</a>     <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \ln \left|\mathbf {X} ^{\top }\mathbf {X} \right|}{\partial \mathbf {X} ^{+}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \ln \left|\mathbf {X} ^{\top }\mathbf {X} \right|}{\partial \mathbf {X} ^{+}}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5346076d93f932ee62e5133106f5d879c553012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.777ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial \ln \left|\mathbf {X} ^{\top }\mathbf {X} \right|}{\partial \mathbf {X} ^{+}}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2\mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2\mathbf {X} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e52b295ee5b26e0e2d26c84ba021e1566baa516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.99ex; height:2.343ex;" alt="{\displaystyle -2\mathbf {X} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2\mathbf {X} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2\mathbf {X} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864800dccab335d0480257dd0d0c45f21541a4cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle -2\mathbf {X} ^{\top }}"> </td></tr> <tr> <td>A is not a function of X, X is square and invertible</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1610acba65481e9b70a28d6bea0d66be77e43fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.858ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\mathbf {X} ^{-1}=2\left|\mathbf {X^{\top }} \right||\mathbf {A} ||\mathbf {X} |\mathbf {X} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\mathbf {X} ^{-1}=2\left|\mathbf {X^{\top }} \right||\mathbf {A} ||\mathbf {X} |\mathbf {X} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd7631d479475f4d20b4e34387002b12a98d4d92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.99ex; height:3.509ex;" alt="{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\mathbf {X} ^{-1}=2\left|\mathbf {X^{\top }} \right||\mathbf {A} ||\mathbf {X} |\mathbf {X} ^{-1}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\left(\mathbf {X} ^{-1}\right)^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\left(\mathbf {X} ^{-1}\right)^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8708cc10b6e974da1939fba5a32c1dd8fe6490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.793ex; height:4.009ex;" alt="{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\left(\mathbf {X} ^{-1}\right)^{\top }}"> </td></tr> <tr> <td>A is not a function of X, X is non-square, A is symmetric</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1610acba65481e9b70a28d6bea0d66be77e43fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.858ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|}{\partial \mathbf {X} }}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}\mathbf {X^{\top }A^{\top }} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">X</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}\mathbf {X^{\top }A^{\top }} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a698eab6629bfa1dda94ee3c489c1ab5e88dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.403ex; height:4.009ex;" alt="{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}\mathbf {X^{\top }A^{\top }} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\mathbf {AX} \left(\mathbf {X^{\top }AX} \right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\mathbf {AX} \left(\mathbf {X^{\top }AX} \right)^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253a3be22f6de1703bd0f6be6bc6fe4c02fc15a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.87ex; height:4.009ex;" alt="{\displaystyle 2\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|\mathbf {AX} \left(\mathbf {X^{\top }AX} \right)^{-1}}"> </td></tr> <tr> <td>A is not a function of X, X is non-square, A is non-symmetric</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial |\mathbf {X^{\top }} \mathbf {A} \mathbf {X} |}{\partial \mathbf {X} }}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial |\mathbf {X^{\top }} \mathbf {A} \mathbf {X} |}{\partial \mathbf {X} }}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c5f36303b92762bf69577a44dfcdda710f799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.47ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial |\mathbf {X^{\top }} \mathbf {A} \mathbf {X} |}{\partial \mathbf {X} }}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|{\Big (}&\left(\mathbf {X^{\top }AX} \right)^{-1}\mathbf {X^{\top }A} +{}\\&\left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}\mathbf {X^{\top }A^{\top }} {\Big )}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> </mtd> <mtd> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">X</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|{\Big (}&\left(\mathbf {X^{\top }AX} \right)^{-1}\mathbf {X^{\top }A} +{}\\&\left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}\mathbf {X^{\top }A^{\top }} {\Big )}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf2748044d93fd709fcb5cd718673d0ab81127d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:33.38ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|{\Big (}&\left(\mathbf {X^{\top }AX} \right)^{-1}\mathbf {X^{\top }A} +{}\\&\left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}\mathbf {X^{\top }A^{\top }} {\Big )}\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|{\Big (}&\mathbf {AX} \left(\mathbf {X^{\top }AX} \right)^{-1}+{}\\&\mathbf {A^{\top }X} \left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}{\Big )}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">X</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">X</mi> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi mathvariant="bold">X</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|{\Big (}&\mathbf {AX} \left(\mathbf {X^{\top }AX} \right)^{-1}+{}\\&\mathbf {A^{\top }X} \left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}{\Big )}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69ecee1b3ca40a862835dc08f98c94555415775e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:31.869ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\left|\mathbf {X^{\top }} \mathbf {A} \mathbf {X} \right|{\Big (}&\mathbf {AX} \left(\mathbf {X^{\top }AX} \right)^{-1}+{}\\&\mathbf {A^{\top }X} \left(\mathbf {X^{\top }A^{\top }X} \right)^{-1}{\Big )}\end{aligned}}}"> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Matrix-by-scalar_identities">Matrix-by-scalar identities</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=22" title="Edit section: Matrix-by-scalar identities">edit</a>]</div> <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Identities: matrix-by-scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565884c84274a792e9b5af680a30f550eaf5e3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}}"> </caption> <tbody><tr> <th scope="col" width="175">Condition </th> <th scope="col" width="100">Expression </th> <th scope="col" width="100">Numerator layout, i.e. by Y </th></tr> <tr> <td>U = U(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial a\mathbf {U} }{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial a\mathbf {U} }{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59afb52f496a4b72e926d1b44a65ce52cc645e59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.894ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial a\mathbf {U} }{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\frac {\partial \mathbf {U} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\frac {\partial \mathbf {U} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbafa5c2f8bf93caa719ddfed605d1ee4f3256a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.441ex; height:5.509ex;" alt="{\displaystyle a{\frac {\partial \mathbf {U} }{\partial x}}}"> </td></tr> <tr> <td>A, B are not functions of x, U = U(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {AUB} }{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">U</mi> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {AUB} }{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ac604cf0639b2d19bd348f2376dbe5919c5b10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.585ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {AUB} }{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} {\frac {\partial \mathbf {U} }{\partial x}}\mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} {\frac {\partial \mathbf {U} }{\partial x}}\mathbf {B} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc4e1d9f7db0d84971013fa3ac701961f950f65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.131ex; height:5.509ex;" alt="{\displaystyle \mathbf {A} {\frac {\partial \mathbf {U} }{\partial x}}\mathbf {B} }"> </td></tr> <tr> <td>U = U(x), V = V(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {U} +\mathbf {V} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {U} +\mathbf {V} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e679a5c864008ff38f52d4ef7d61e3e4c8aa30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.333ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {U} +\mathbf {V} )}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {U} }{\partial x}}+{\frac {\partial \mathbf {V} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {U} }{\partial x}}+{\frac {\partial \mathbf {V} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c888bad95263e09dedcae7b843aef60230a019ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.225ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {U} }{\partial x}}+{\frac {\partial \mathbf {V} }{\partial x}}}"> </td></tr> <tr> <td>U = U(x), V = V(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {U} \mathbf {V} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {U} \mathbf {V} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/872a42c30ceb41fcee8b176f03d1c215048533d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.493ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {U} \mathbf {V} )}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {V} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f4ea5b61d97d9976789433a5dfaba80244fd69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.301ex; height:5.509ex;" alt="{\displaystyle \mathbf {U} {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {V} }"> </td></tr> <tr> <td>U = U(x), V = V(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {U} \otimes \mathbf {V} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {U} \otimes \mathbf {V} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8850ae62fa9fb9a4def09235dc033473fa3fee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.333ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {U} \otimes \mathbf {V} )}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} \otimes {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\otimes \mathbf {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} \otimes {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\otimes \mathbf {V} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d591fd4a1a65f880f6f192c8a333dd5a18da9f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.982ex; height:5.509ex;" alt="{\displaystyle \mathbf {U} \otimes {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\otimes \mathbf {V} }"> </td></tr> <tr> <td>U = U(x), V = V(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {U} \circ \mathbf {V} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {U} \circ \mathbf {V} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd85c5060feefe7503be10d610742f072497f6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.688ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {U} \circ \mathbf {V} )}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} \circ {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\circ \mathbf {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} \circ {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\circ \mathbf {V} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525fb93f72a2f760a77092cb990242162262811c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.691ex; height:5.509ex;" alt="{\displaystyle \mathbf {U} \circ {\frac {\partial \mathbf {V} }{\partial x}}+{\frac {\partial \mathbf {U} }{\partial x}}\circ \mathbf {V} }"> </td></tr> <tr> <td>U = U(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {U} ^{-1}}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {U} ^{-1}}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa138aa865ede81c2fc47f61f5d09a1f1962265d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.997ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {U} ^{-1}}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e2ed7bc4a1448d2bb78c5fb9b8f51d28312bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.798ex; height:5.509ex;" alt="{\displaystyle -\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}}"> </td></tr> <tr> <td>U = U(x,y)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\mathbf {U} ^{-1}}{\partial x\partial y}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\mathbf {U} ^{-1}}{\partial x\partial y}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/063994fb9bfb205e94432f8aaacec5bec3154d97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.076ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}\mathbf {U} ^{-1}}{\partial x\partial y}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} ^{-1}\left({\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial y}}-{\frac {\partial ^{2}\mathbf {U} }{\partial x\partial y}}+{\frac {\partial \mathbf {U} }{\partial y}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)\mathbf {U} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} ^{-1}\left({\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial y}}-{\frac {\partial ^{2}\mathbf {U} }{\partial x\partial y}}+{\frac {\partial \mathbf {U} }{\partial y}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)\mathbf {U} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90e8e807c09105537ee4ff1e12832f81208a0d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.235ex; height:6.343ex;" alt="{\displaystyle \mathbf {U} ^{-1}\left({\frac {\partial \mathbf {U} }{\partial x}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial y}}-{\frac {\partial ^{2}\mathbf {U} }{\partial x\partial y}}+{\frac {\partial \mathbf {U} }{\partial y}}\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)\mathbf {U} ^{-1}}"> </td></tr> <tr> <td>A is not a function of x, g(X) is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. eX, sin(X), cos(X), ln(X), etc.); g(x) is the equivalent scalar function, g<big>′</big>(x) is its derivative, and g<big>′</big>(X) is the corresponding matrix function</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \,\mathbf {g} (x\mathbf {A} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \,\mathbf {g} (x\mathbf {A} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afded1249ac0bab5521dcde2d48a52cdecdbbeb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.49ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \,\mathbf {g} (x\mathbf {A} )}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \mathbf {g} '(x\mathbf {A} )=\mathbf {g} '(x\mathbf {A} )\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \mathbf {g} '(x\mathbf {A} )=\mathbf {g} '(x\mathbf {A} )\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee1740d13819db72124adf5f480dbea96b13a3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.497ex; height:3.009ex;" alt="{\displaystyle \mathbf {A} \mathbf {g} '(x\mathbf {A} )=\mathbf {g} '(x\mathbf {A} )\mathbf {A} }"> </td></tr> <tr> <td>A is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial e^{x\mathbf {A} }}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial e^{x\mathbf {A} }}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00e65d9a11cfd101815c5b2503117ef3a6fc091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.291ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial e^{x\mathbf {A} }}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} e^{x\mathbf {A} }=e^{x\mathbf {A} }\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} e^{x\mathbf {A} }=e^{x\mathbf {A} }\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca2c46d5288ce58fdee3237ccdb9c50302edc9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.505ex; height:2.676ex;" alt="{\displaystyle \mathbf {A} e^{x\mathbf {A} }=e^{x\mathbf {A} }\mathbf {A} }"> </td></tr></tbody></table></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Derivative_of_the_exponential_map" title="Derivative of the exponential map">Derivative of the exponential map</a></div> <div class="mw-heading mw-heading3"><h3 id="Scalar-by-scalar_identities">Scalar-by-scalar identities</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=23" title="Edit section: Scalar-by-scalar identities">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="With_vectors_involved">With vectors involved</h4>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=24" title="Edit section: With vectors involved">edit</a>]</div> <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Identities: scalar-by-scalar, with vectors involved </caption> <tbody><tr> <th scope="col" width="150">Condition </th> <th scope="col" width="10">Expression </th> <th scope="col" width="150">Any layout (assumes <a href="/wiki/Dot_product" title="Dot product">dot product</a> ignores row vs. column layout) </th></tr> <tr> <td>u = u(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial g(\mathbf {u} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g(\mathbf {u} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba5612262dd53992b0a461de4225d5e400aa284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.018ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial g(\mathbf {u} )}{\partial x}}=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial g(\mathbf {u} )}{\partial \mathbf {u} }}\cdot {\frac {\partial \mathbf {u} }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g(\mathbf {u} )}{\partial \mathbf {u} }}\cdot {\frac {\partial \mathbf {u} }{\partial x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775144508441860cb06c4f337028ae480acce3bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.883ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial g(\mathbf {u} )}{\partial \mathbf {u} }}\cdot {\frac {\partial \mathbf {u} }{\partial x}}}"> </td></tr> <tr> <td>u = u(x), v = v(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eec170ee474510e3aa63c8f4b083d680ca8a6c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.992ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (\mathbf {u} \cdot \mathbf {v} )}{\partial x}}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \cdot {\frac {\partial \mathbf {v} }{\partial x}}+{\frac {\partial \mathbf {u} }{\partial x}}\cdot \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot {\frac {\partial \mathbf {v} }{\partial x}}+{\frac {\partial \mathbf {u} }{\partial x}}\cdot \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f5a2b0dcc2881418f7c2635ff72bf2b52e4eaec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.3ex; height:5.509ex;" alt="{\displaystyle \mathbf {u} \cdot {\frac {\partial \mathbf {v} }{\partial x}}+{\frac {\partial \mathbf {u} }{\partial x}}\cdot \mathbf {v} }"> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading4"><h4 id="With_matrices_involved">With matrices involved</h4>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=25" title="Edit section: With matrices involved">edit</a>]</div> <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Identities: scalar-by-scalar, with matrices involved<a href="#cite_note-matrix-cookbook-4">[3]</a> </caption> <tbody><tr> <th scope="col" width="175">Condition </th> <th scope="col" width="100">Expression </th> <th scope="col" width="100">Consistent numerator layout, i.e. by Y and XT </th> <th scope="col" width="100">Mixed layout, i.e. by Y and X </th></tr> <tr> <td>U = U(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial |\mathbf {U} |}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial |\mathbf {U} |}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cecc7cf6dde87f15797a804ffb7ada25d0422f41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.958ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial |\mathbf {U} |}{\partial x}}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {U} |\operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {U} |\operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13074d057948a7b9988de79daccb1f7b2305ede1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.575ex; height:6.176ex;" alt="{\displaystyle |\mathbf {U} |\operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)}"> </td></tr> <tr> <td>U = U(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \ln |\mathbf {U} |}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \ln |\mathbf {U} |}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87e0c05e2145e1facdbbdfbdda0db812ff8b6ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.671ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \ln |\mathbf {U} |}{\partial x}}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6f7eea75b36fa7e2fb87fa3122672300bf4056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.838ex; height:6.176ex;" alt="{\displaystyle \operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)}"> </td></tr> <tr> <td>U = U(x)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}|\mathbf {U} |}{\partial x^{2}}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}|\mathbf {U} |}{\partial x^{2}}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b628e5cf453bb8d92b011e0910460e2a5eee6819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.037ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial ^{2}|\mathbf {U} |}{\partial x^{2}}}=}"> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\mathbf {U} \right|\left[\operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial ^{2}\mathbf {U} }{\partial x^{2}}}\right)+\operatorname {tr} ^{2}\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)-\operatorname {tr} \left(\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)^{2}\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo>|</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>tr</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\mathbf {U} \right|\left[\operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial ^{2}\mathbf {U} }{\partial x^{2}}}\right)+\operatorname {tr} ^{2}\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)-\operatorname {tr} \left(\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)^{2}\right)\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16c46255b081a2f4f645e0cbac5b913a1537a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.511ex; height:7.509ex;" alt="{\displaystyle \left|\mathbf {U} \right|\left[\operatorname {tr} \left(\mathbf {U} ^{-1}{\frac {\partial ^{2}\mathbf {U} }{\partial x^{2}}}\right)+\operatorname {tr} ^{2}\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)-\operatorname {tr} \left(\left(\mathbf {U} ^{-1}{\frac {\partial \mathbf {U} }{\partial x}}\right)^{2}\right)\right]}"> </td></tr> <tr> <td>U = U(x) </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial g(\mathbf {U} )}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial g(\mathbf {U} )}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1acb3d37a01fd2a62f2f0ba91e873b9ef050cb7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.589ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial g(\mathbf {U} )}{\partial x}}=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial x}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7ef3caf2f94ba324bcce6692e7ba9a041a56e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.584ex; height:6.343ex;" alt="{\displaystyle \operatorname {tr} \left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}{\frac {\partial \mathbf {U} }{\partial x}}\right)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\top }{\frac {\partial \mathbf {U} }{\partial x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\top }{\frac {\partial \mathbf {U} }{\partial x}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a63cd62d1b8e58adf356c37c03b42255237bc326" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.777ex; height:7.509ex;" alt="{\displaystyle \operatorname {tr} \left(\left({\frac {\partial g(\mathbf {U} )}{\partial \mathbf {U} }}\right)^{\top }{\frac {\partial \mathbf {U} }{\partial x}}\right)}"> </td></tr> <tr> <td>A is not a function of x, g(X) is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. eX, sin(X), cos(X), ln(X), etc.); g(x) is the equivalent scalar function, g<big>′</big>(x) is its derivative, and g<big>′</big>(X) is the corresponding matrix function.</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {g} (x\mathbf {A} ))}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {g} (x\mathbf {A} ))}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b63babd3d34a99e8e92ec870b4966a02f9a822ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.115ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} (\mathbf {g} (x\mathbf {A} ))}{\partial x}}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\mathbf {A} \mathbf {g} '(x\mathbf {A} )\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\mathbf {A} \mathbf {g} '(x\mathbf {A} )\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2bc17790e477e66bc277cef545a74dc89122d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.825ex; height:3.009ex;" alt="{\displaystyle \operatorname {tr} \left(\mathbf {A} \mathbf {g} '(x\mathbf {A} )\right)}"> </td></tr> <tr> <td>A is not a function of x</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \operatorname {tr} \left(e^{x\mathbf {A} }\right)}{\partial x}}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \operatorname {tr} \left(e^{x\mathbf {A} }\right)}{\partial x}}=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8007ebc0e98d66201a60d817f7ff85d55c3a33d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.625ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial \operatorname {tr} \left(e^{x\mathbf {A} }\right)}{\partial x}}=}"></td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\mathbf {A} e^{x\mathbf {A} }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\mathbf {A} e^{x\mathbf {A} }\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ec4a31f2d1b00980735e1bf28e357bc3ef5eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.65ex; height:3.343ex;" alt="{\displaystyle \operatorname {tr} \left(\mathbf {A} e^{x\mathbf {A} }\right)}"> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Identities_in_differential_form">Identities in differential form</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=26" title="Edit section: Identities in differential form">edit</a>]</div> It is often easier to work in differential form and then convert back to normal derivatives. This only works well using the numerator layout. In these rules, a is a scalar. <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Differential identities: scalar involving matrix<a href="#cite_note-minka-1">[1]</a><a href="#cite_note-matrix-cookbook-4">[3]</a> </caption> <tbody><tr> <th>Expression</th> <th>Result (numerator layout) </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\operatorname {tr} (\mathbf {X} ))=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\operatorname {tr} (\mathbf {X} ))=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5dc91e24553a1473e6cec689c561fcdacf589b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.124ex; height:2.843ex;" alt="{\displaystyle d(\operatorname {tr} (\mathbf {X} ))=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (d\mathbf {X} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (d\mathbf {X} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234412411bd5861af0f17b144a7238b5cf9707e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.861ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (d\mathbf {X} )}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(|\mathbf {X} |)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(|\mathbf {X} |)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/967384df794437f3b37679aa8e82167b646a1773" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.792ex; height:2.843ex;" alt="{\displaystyle d(|\mathbf {X} |)=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {X} |\operatorname {tr} \left(\mathbf {X} ^{-1}d\mathbf {X} \right)=\operatorname {tr} (\operatorname {adj} (\mathbf {X} )d\mathbf {X} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {X} |\operatorname {tr} \left(\mathbf {X} ^{-1}d\mathbf {X} \right)=\operatorname {tr} (\operatorname {adj} (\mathbf {X} )d\mathbf {X} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd15904de92b2329944348c6e5a70ed9589e927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.189ex; height:3.343ex;" alt="{\displaystyle |\mathbf {X} |\operatorname {tr} \left(\mathbf {X} ^{-1}d\mathbf {X} \right)=\operatorname {tr} (\operatorname {adj} (\mathbf {X} )d\mathbf {X} )}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\ln |\mathbf {X} |)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\ln |\mathbf {X} |)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3259d0f3b205654dd6cb94d14333a2255869a95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.118ex; height:2.843ex;" alt="{\displaystyle d(\ln |\mathbf {X} |)=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\mathbf {X} ^{-1}d\mathbf {X} \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\mathbf {X} ^{-1}d\mathbf {X} \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c70f56b35822017d5a03d5ddff897658c56c4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.534ex; height:3.343ex;" alt="{\displaystyle \operatorname {tr} \left(\mathbf {X} ^{-1}d\mathbf {X} \right)}"> </td></tr></tbody></table></dd></dl> <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Differential identities: matrix<a href="#cite_note-minka-1">[1]</a><a href="#cite_note-matrix-cookbook-4">[3]</a><a href="#cite_note-9">[6]</a> <a href="#cite_note-10">[7]</a> </caption> <tbody><tr> <th>Condition</th> <th>Expression</th> <th>Result (numerator layout) </th></tr> <tr> <td>A is not a function of X</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathbf {A} )=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\mathbf {A} )=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7e6e2ff84cff62eace2f5d1fee6ca89a2938ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.498ex; height:2.843ex;" alt="{\displaystyle d(\mathbf {A} )=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> </td></tr> <tr> <td>a is not a function of X</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(a\mathbf {X} )=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(a\mathbf {X} )=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91c101b0891c47b3f1d20d96b141125474867ae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.728ex; height:2.843ex;" alt="{\displaystyle d(a\mathbf {X} )=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,d\mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,d\mathbf {X} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c3d97e20baf0ea9790cd63e7207128846b38d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.852ex; height:2.176ex;" alt="{\displaystyle a\,d\mathbf {X} }"> </td></tr> <tr> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathbf {X} +\mathbf {Y} )=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\mathbf {X} +\mathbf {Y} )=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b73dba0b049337a0cef941d976498d1dabe4bc76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.358ex; height:2.843ex;" alt="{\displaystyle d(\mathbf {X} +\mathbf {Y} )=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {X} +d\mathbf {Y} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {X} +d\mathbf {Y} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30ac4c458061ea58a52b5a18e676bc2a7f1e1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.311ex; height:2.343ex;" alt="{\displaystyle d\mathbf {X} +d\mathbf {Y} }"> </td></tr> <tr> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathbf {X} \mathbf {Y} )=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\mathbf {X} \mathbf {Y} )=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83613591465e8a82e10550e0e8b3e1e80b88a07e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.517ex; height:2.843ex;" alt="{\displaystyle d(\mathbf {X} \mathbf {Y} )=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d\mathbf {X} )\mathbf {Y} +\mathbf {X} (d\mathbf {Y} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d\mathbf {X} )\mathbf {Y} +\mathbf {X} (d\mathbf {Y} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9909db6929624a5a28fb860e7eadfc52a98ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.969ex; height:2.843ex;" alt="{\displaystyle (d\mathbf {X} )\mathbf {Y} +\mathbf {X} (d\mathbf {Y} )}"> </td></tr> <tr> <td>(<a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a>)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathbf {X} \otimes \mathbf {Y} )=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\mathbf {X} \otimes \mathbf {Y} )=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f02f0e65f7a307f516bc337a7e313dac84d306c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.358ex; height:2.843ex;" alt="{\displaystyle d(\mathbf {X} \otimes \mathbf {Y} )=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d\mathbf {X} )\otimes \mathbf {Y} +\mathbf {X} \otimes (d\mathbf {Y} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d\mathbf {X} )\otimes \mathbf {Y} +\mathbf {X} \otimes (d\mathbf {Y} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/061faf5c948446b1ef62e6a46f5dccf2f1ce2286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.65ex; height:2.843ex;" alt="{\displaystyle (d\mathbf {X} )\otimes \mathbf {Y} +\mathbf {X} \otimes (d\mathbf {Y} )}"> </td></tr> <tr> <td>(<a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a>)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\mathbf {X} \circ \mathbf {Y} )=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\mathbf {X} \circ \mathbf {Y} )=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271eaffb1ef30b89a299e00da5444cc13402e2c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.712ex; height:2.843ex;" alt="{\displaystyle d(\mathbf {X} \circ \mathbf {Y} )=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d\mathbf {X} )\circ \mathbf {Y} +\mathbf {X} \circ (d\mathbf {Y} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>∘</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d\mathbf {X} )\circ \mathbf {Y} +\mathbf {X} \circ (d\mathbf {Y} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1c9d41a3d0a7e2c37cdacc9587e3b12c72e742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.358ex; height:2.843ex;" alt="{\displaystyle (d\mathbf {X} )\circ \mathbf {Y} +\mathbf {X} \circ (d\mathbf {Y} )}"> </td></tr> <tr> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\left(\mathbf {X} ^{\top }\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(\mathbf {X} ^{\top }\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80e08fac2a1857d6eeeca6e6937a62eb969dbd77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.716ex; height:3.343ex;" alt="{\displaystyle d\left(\mathbf {X} ^{\top }\right)=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d\mathbf {X} )^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d\mathbf {X} )^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2d0c0a81802de0c8588dd215bbd3b2c2eaf675" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.555ex; height:3.176ex;" alt="{\displaystyle (d\mathbf {X} )^{\top }}"> </td></tr> <tr> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\left(\mathbf {X} ^{-1}\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(\mathbf {X} ^{-1}\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/221625109e14bae9bde2f8e4642dc8b35c03f8ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.538ex; height:3.343ex;" alt="{\displaystyle d\left(\mathbf {X} ^{-1}\right)=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mathbf {X} ^{-1}\left(d\mathbf {X} \right)\mathbf {X} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mathbf {X} ^{-1}\left(d\mathbf {X} \right)\mathbf {X} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12a8f577a94f9713686bf68d414c0a87f244a85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.332ex; height:3.176ex;" alt="{\displaystyle -\mathbf {X} ^{-1}\left(d\mathbf {X} \right)\mathbf {X} ^{-1}}"> </td></tr> <tr> <td>(<a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>)</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\left(\mathbf {X} ^{\mathrm {H} }\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(\mathbf {X} ^{\mathrm {H} }\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2b3a30a01c0ec5ce1f3943490983e25e6c2bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.67ex; height:3.343ex;" alt="{\displaystyle d\left(\mathbf {X} ^{\mathrm {H} }\right)=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d\mathbf {X} )^{\mathrm {H} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d\mathbf {X} )^{\mathrm {H} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53352dab45f08687d10121ef89c37e4751290c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.509ex; height:3.176ex;" alt="{\displaystyle (d\mathbf {X} )^{\mathrm {H} }}"> </td></tr> <tr> <td>n is a positive integer</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\left(\mathbf {X} ^{n}\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(\mathbf {X} ^{n}\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/138f34ef733c829542c0bc43eafe83735caf69be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.104ex; height:2.843ex;" alt="{\displaystyle d\left(\mathbf {X} ^{n}\right)=}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n-1}\mathbf {X} ^{i}(d\mathbf {X} )\mathbf {X} ^{n-i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n-1}\mathbf {X} ^{i}(d\mathbf {X} )\mathbf {X} ^{n-i-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d5be2d2973b96349e2dd29ef0793cedf1057d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.79ex; height:7.343ex;" alt="{\displaystyle \sum _{i=0}^{n-1}\mathbf {X} ^{i}(d\mathbf {X} )\mathbf {X} ^{n-i-1}}"> </td></tr> <tr> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\left(e^{\mathbf {X} }\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(e^{\mathbf {X} }\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9408091b5f6ff868ace9e364b53fc4727507a90f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.93ex; height:3.343ex;" alt="{\displaystyle d\left(e^{\mathbf {X} }\right)=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}e^{a\mathbf {X} }(d\mathbf {X} )e^{(1-a)\mathbf {X} }\,da}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}e^{a\mathbf {X} }(d\mathbf {X} )e^{(1-a)\mathbf {X} }\,da}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ffe1913e349e08b1de8395c9f6e33ad7eaf188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.39ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}e^{a\mathbf {X} }(d\mathbf {X} )e^{(1-a)\mathbf {X} }\,da}"> </td></tr> <tr> <td> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\left(\log {X}\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(\log {X}\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8c79a16cd1efcde97a56d053ed47c273e35b69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.204ex; height:2.843ex;" alt="{\displaystyle d\left(\log {X}\right)=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\infty }(\mathbf {X} +z\,\mathbf {I} )^{-1}(d\mathbf {X} )(\mathbf {X} +z\,\mathbf {I} )^{-1}\,dz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>+</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>+</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }(\mathbf {X} +z\,\mathbf {I} )^{-1}(d\mathbf {X} )(\mathbf {X} +z\,\mathbf {I} )^{-1}\,dz}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c21308a4c9b0710011d7ba4a3e1538105ad8eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.057ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }(\mathbf {X} +z\,\mathbf {I} )^{-1}(d\mathbf {X} )(\mathbf {X} +z\,\mathbf {I} )^{-1}\,dz}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =\sum _{i}\lambda _{i}\mathbf {P} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =\sum _{i}\lambda _{i}\mathbf {P} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaacf81b2c42120d4cb051e23e9ae7a15b321990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.641ex; height:5.509ex;" alt="{\displaystyle \mathbf {X} =\sum _{i}\lambda _{i}\mathbf {P} _{i}}"> is <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">diagonalizable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{i}\mathbf {P} _{j}=\delta _{ij}\mathbf {P} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{i}\mathbf {P} _{j}=\delta _{ij}\mathbf {P} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9eb96d2cd3e73ba3599e91caf614f84cb184f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.597ex; height:3.009ex;" alt="{\displaystyle \mathbf {P} _{i}\mathbf {P} _{j}=\delta _{ij}\mathbf {P} _{i}}"> f is <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> at every eigenvalue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\left(f(\mathbf {X} )\right)=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\left(f(\mathbf {X} )\right)=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43fb89331bfa09608f7421dba03bb7d6795f8e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.973ex; height:2.843ex;" alt="{\displaystyle d\left(f(\mathbf {X} )\right)=}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{ij}\mathbf {P} _{i}(d\mathbf {X} )\mathbf {P} _{j}{\begin{cases}f'(\lambda _{i})&\lambda _{i}=\lambda _{j}\\{\frac {f(\lambda _{i})-f(\lambda _{j})}{\lambda _{i}-\lambda _{j}}}&\lambda _{i}\neq \lambda _{j}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{ij}\mathbf {P} _{i}(d\mathbf {X} )\mathbf {P} _{j}{\begin{cases}f'(\lambda _{i})&\lambda _{i}=\lambda _{j}\\{\frac {f(\lambda _{i})-f(\lambda _{j})}{\lambda _{i}-\lambda _{j}}}&\lambda _{i}\neq \lambda _{j}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb2e7ba1571c86d1d5dad8739fc0ae905f3629b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.352ex; height:7.843ex;" alt="{\displaystyle \sum _{ij}\mathbf {P} _{i}(d\mathbf {X} )\mathbf {P} _{j}{\begin{cases}f'(\lambda _{i})&\lambda _{i}=\lambda _{j}\\{\frac {f(\lambda _{i})-f(\lambda _{j})}{\lambda _{i}-\lambda _{j}}}&\lambda _{i}\neq \lambda _{j}\end{cases}}}"> </td></tr></tbody></table></dd></dl> In the last row, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa75d04c11480d976e1396951e02cbb3c4f71568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}"> is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {P} _{k})_{ij}=(\mathbf {Q} )_{ik}(\mathbf {Q} ^{-1})_{kj}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {P} _{k})_{ij}=(\mathbf {Q} )_{ik}(\mathbf {Q} ^{-1})_{kj}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aff897833891001536ee7f9ce1d8920e93d8445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.69ex; height:3.343ex;" alt="{\displaystyle (\mathbf {P} _{k})_{ij}=(\mathbf {Q} )_{ik}(\mathbf {Q} ^{-1})_{kj}}"> is the set of orthogonal projection operators that project onto the k-th eigenvector of X. Q is the matrix of <a href="/wiki/Eigendecomposition_of_a_matrix#Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">eigenvectors</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Λ</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60ce9ed670e96dfdfc76859202fe87f2567b309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.34ex; height:3.009ex;" alt="{\displaystyle \mathbf {X} =\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{-1}}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\boldsymbol {\Lambda }})_{ii}=\lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Λ</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\Lambda }})_{ii}=\lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce404fcc468cefa84dc24ec646077c296d61020b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.303ex; height:2.843ex;" alt="{\displaystyle ({\boldsymbol {\Lambda }})_{ii}=\lambda _{i}}"> are the eigenvalues. The matrix function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {X} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {X} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07ed90d175ff609309c2a46bb05b75fc43f4000a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.107ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {X} )}"> is <a href="/wiki/Eigendecomposition_of_a_matrix#Functional_calculus" title="Eigendecomposition of a matrix">defined in terms of the scalar function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> for diagonalizable matrices by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(\mathbf {X} )=\sum _{i}f(\lambda _{i})\mathbf {P} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(\mathbf {X} )=\sum _{i}f(\lambda _{i})\mathbf {P} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a263daa33a865cac97227c8889a803321fc88dcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.716ex; height:3.009ex;" alt="{\textstyle f(\mathbf {X} )=\sum _{i}f(\lambda _{i})\mathbf {P} _{i}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {X} =\sum _{i}\lambda _{i}\mathbf {P} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {X} =\sum _{i}\lambda _{i}\mathbf {P} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d17a45e08f1adf55bd73c0848046e1398b3b6b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.54ex; height:3.009ex;" alt="{\textstyle \mathbf {X} =\sum _{i}\lambda _{i}\mathbf {P} _{i}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{i}\mathbf {P} _{j}=\delta _{ij}\mathbf {P} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{i}\mathbf {P} _{j}=\delta _{ij}\mathbf {P} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9eb96d2cd3e73ba3599e91caf614f84cb184f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.597ex; height:3.009ex;" alt="{\displaystyle \mathbf {P} _{i}\mathbf {P} _{j}=\delta _{ij}\mathbf {P} _{i}}">. To convert to normal derivative form, first convert it to one of the following canonical forms, and then use these identities: <dl><dd><table class="wikitable" style="text-align: center;"> <caption>Conversion from differential to derivative form<a href="#cite_note-minka-1">[1]</a> </caption> <tbody><tr> <th>Canonical differential form</th> <th>Equivalent derivative form (numerator layout) </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy=a\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy=a\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce456dc40793632c19c009d6170c74ae644f784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.632ex; height:2.509ex;" alt="{\displaystyle dy=a\,dx}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{dx}}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{dx}}=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e758428a1022503fdaeffc5e287c5bf7bbdc06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.71ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{dx}}=a}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy=\mathbf {a} ^{\top }d\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy=\mathbf {a} ^{\top }d\mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3582c2ebb46ffcc7f60192f15eb4d97a28d8774" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.907ex; height:3.009ex;" alt="{\displaystyle dy=\mathbf {a} ^{\top }d\mathbf {x} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{d\mathbf {x} }}=\mathbf {a} ^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{d\mathbf {x} }}=\mathbf {a} ^{\top }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5600fc3e98a1e1d3c28915bdf56e07a22588d7a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.372ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{d\mathbf {x} }}=\mathbf {a} ^{\top }}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy=\operatorname {tr} (\mathbf {A} \,d\mathbf {X} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy=\operatorname {tr} (\mathbf {A} \,d\mathbf {X} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a33eead0af35733e6ac1ca42f6c899731b531eed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.737ex; height:2.843ex;" alt="{\displaystyle dy=\operatorname {tr} (\mathbf {A} \,d\mathbf {X} )}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dy}{d\mathbf {X} }}=\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dy}{d\mathbf {X} }}=\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8d6694e983da55260de8ef14f1bf7bdb85b40f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.189ex; height:5.509ex;" alt="{\displaystyle {\frac {dy}{d\mathbf {X} }}=\mathbf {A} }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {y} =\mathbf {a} \,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {y} =\mathbf {a} \,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0ae3cf15e1267f4fdb141e1d68349927727bef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.957ex; height:2.509ex;" alt="{\displaystyle d\mathbf {y} =\mathbf {a} \,dx}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {y} }{dx}}=\mathbf {a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {y} }{dx}}=\mathbf {a} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faa54d96218197dadd7b4f1e3471443c0b79a6d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.861ex; height:5.509ex;" alt="{\displaystyle {\frac {d\mathbf {y} }{dx}}=\mathbf {a} }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {y} =\mathbf {A} \,d\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {y} =\mathbf {A} \,d\mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e99d558f59946cfc8d0cb3b995a49c42fead04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.759ex; height:2.509ex;" alt="{\displaystyle d\mathbf {y} =\mathbf {A} \,d\mathbf {x} }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {y} }{d\mathbf {x} }}=\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {y} }{d\mathbf {x} }}=\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5649f8fff55607c21a17cdb42c9c5d15b2f19b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.581ex; height:5.509ex;" alt="{\displaystyle {\frac {d\mathbf {y} }{d\mathbf {x} }}=\mathbf {A} }"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {Y} =\mathbf {A} \,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {Y} =\mathbf {A} \,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a371dad51f295c2fee9df1952f2bdfaa856c254e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.286ex; height:2.176ex;" alt="{\displaystyle d\mathbf {Y} =\mathbf {A} \,dx}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\mathbf {Y} }{dx}}=\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {Y} }{dx}}=\mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c48de4dccee87aaf5fba393272c94b5a8065af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.189ex; height:5.509ex;" alt="{\displaystyle {\frac {d\mathbf {Y} }{dx}}=\mathbf {A} }"> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=27" title="Edit section: Applications">edit</a>]</div> Matrix differential calculus is used in statistics and econometrics, particularly for the statistical analysis of <a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">multivariate distributions</a>, especially the <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a> and other <a href="/wiki/Elliptical_distribution" title="Elliptical distribution">elliptical distributions</a>.<a href="#cite_note-11">[8]</a><a href="#cite_note-12">[9]</a><a href="#cite_note-13">[10]</a> It is used in <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a> to compute, for example, the <a href="/wiki/Linear_least_squares_(mathematics)#The_general_problem" class="mw-redirect" title="Linear least squares (mathematics)">ordinary least squares regression formula</a> for the case of multiple <a href="/wiki/Explanatory_variable" class="mw-redirect" title="Explanatory variable">explanatory variables</a>.<a href="#cite_note-14">[11]</a> It is also used in random matrices, statistical moments, local sensitivity and statistical diagnostics.<a href="#cite_note-15">[12]</a> <a href="#cite_note-16">[13]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=28" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> <ul><li><a href="/wiki/Derivative_(generalizations)" class="mw-redirect" title="Derivative (generalizations)">Derivative (generalizations)</a></li> <li><a href="/wiki/Product_integral" title="Product integral">Product integral</a></li> <li><a href="/wiki/Ricci_calculus#Differentiation" title="Ricci calculus">Ricci calculus</a></li> <li><a href="/wiki/Tensor_derivative" class="mw-redirect" title="Tensor derivative">Tensor derivative</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=29" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-zerovec-3">^ <a href="#cite_ref-zerovec_3-0">a</a> <a href="#cite_ref-zerovec_3-1">b</a> <a href="#cite_ref-zerovec_3-2">c</a> Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"> refers to a <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a> of all 0's, of size n, where n is the length of x. </li> <li id="cite_note-zeromatrix-5">^ <a href="#cite_ref-zeromatrix_5-0">a</a> <a href="#cite_ref-zeromatrix_5-1">b</a> Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }"> refers to a matrix of all 0's, of the same shape as X. </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> The constant a disappears in the result. This is intentional. In general, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\ln au}{dx}}={\frac {1}{au}}{\frac {d(au)}{dx}}={\frac {1}{au}}a{\frac {du}{dx}}={\frac {1}{u}}{\frac {du}{dx}}={\frac {d\ln u}{dx}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mi>u</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\ln au}{dx}}={\frac {1}{au}}{\frac {d(au)}{dx}}={\frac {1}{au}}a{\frac {du}{dx}}={\frac {1}{u}}{\frac {du}{dx}}={\frac {d\ln u}{dx}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87cd9d0312552e0ca10fa857a5573e43dec0194d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:49.832ex; height:5.843ex;" alt="{\displaystyle {\frac {d\ln au}{dx}}={\frac {1}{au}}{\frac {d(au)}{dx}}={\frac {1}{au}}a{\frac {du}{dx}}={\frac {1}{u}}{\frac {du}{dx}}={\frac {d\ln u}{dx}}.}"> or, also <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\ln au}{dx}}={\frac {d(\ln a+\ln u)}{dx}}={\frac {d\ln a}{dx}}+{\frac {d\ln u}{dx}}={\frac {d\ln u}{dx}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ln</mi> <mo>⁡</mo> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\ln au}{dx}}={\frac {d(\ln a+\ln u)}{dx}}={\frac {d\ln a}{dx}}+{\frac {d\ln u}{dx}}={\frac {d\ln u}{dx}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc8c4b642035ffb3113a150cddf7d9a369a2ac3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.208ex; height:5.843ex;" alt="{\displaystyle {\frac {d\ln au}{dx}}={\frac {d(\ln a+\ln u)}{dx}}={\frac {d\ln a}{dx}}+{\frac {d\ln u}{dx}}={\frac {d\ln u}{dx}}.}"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=30" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-minka-1">^ <a href="#cite_ref-minka_1-0">a</a> <a href="#cite_ref-minka_1-1">b</a> <a href="#cite_ref-minka_1-2">c</a> <a href="#cite_ref-minka_1-3">d</a> <a href="#cite_ref-minka_1-4">e</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFThomas_P.2000" class="citation web cs1">Thomas P., Minka (December 28, 2000). <a rel="nofollow" class="external text" href="http://research.microsoft.com/en-us/um/people/minka/papers/matrix/">"Old and New Matrix Algebra Useful for Statistics"</a>. MIT Media Lab note (1997; revised 12/00). Retrieved 5 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Old+and+New+Matrix+Algebra+Useful+for+Statistics&rft.date=2000-12-28&rft.aulast=Thomas+P.&rft.aufirst=Minka&rft_id=http%3A%2F%2Fresearch.microsoft.com%2Fen-us%2Fum%2Fpeople%2Fminka%2Fpapers%2Fmatrix%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFelippa" class="citation book cs1">Felippa, Carlos A. <a rel="nofollow" class="external text" href="http://www.colorado.edu/engineering/cas/courses.d/IFEM.d/IFEM.AppD.d/IFEM.AppD.pdf">"Appendix D, Linear Algebra: Determinants, Inverses, Rank"</a> (PDF). ASEN 5007: Introduction To Finite Element Methods. Boulder, Colorado: University of Colorado. Retrieved 5 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Appendix+D%2C+Linear+Algebra%3A+Determinants%2C+Inverses%2C+Rank&rft.btitle=ASEN+5007%3A+Introduction+To+Finite+Element+Methods&rft.place=Boulder%2C+Colorado&rft.pub=University+of+Colorado&rft.aulast=Felippa&rft.aufirst=Carlos+A.&rft_id=http%3A%2F%2Fwww.colorado.edu%2Fengineering%2Fcas%2Fcourses.d%2FIFEM.d%2FIFEM.AppD.d%2FIFEM.AppD.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> Uses the <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a> (<a href="/wiki/Transpose" title="Transpose">transpose</a> to <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a>) definition of vector and matrix derivatives. </li> <li id="cite_note-matrix-cookbook-4">^ <a href="#cite_ref-matrix-cookbook_4-0">a</a> <a href="#cite_ref-matrix-cookbook_4-1">b</a> <a href="#cite_ref-matrix-cookbook_4-2">c</a> <a href="#cite_ref-matrix-cookbook_4-3">d</a> <a href="#cite_ref-matrix-cookbook_4-4">e</a> <a href="#cite_ref-matrix-cookbook_4-5">f</a> <a href="#cite_ref-matrix-cookbook_4-6">g</a> <a href="#cite_ref-matrix-cookbook_4-7">h</a> <a href="#cite_ref-matrix-cookbook_4-8">i</a> <a href="#cite_ref-matrix-cookbook_4-9">j</a> <a href="#cite_ref-matrix-cookbook_4-10">k</a> <a href="#cite_ref-matrix-cookbook_4-11">l</a> <a href="#cite_ref-matrix-cookbook_4-12">m</a> <a href="#cite_ref-matrix-cookbook_4-13">n</a> <a href="#cite_ref-matrix-cookbook_4-14">o</a> <a href="#cite_ref-matrix-cookbook_4-15">p</a> <a href="#cite_ref-matrix-cookbook_4-16">q</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetersenPedersen" class="citation book cs1">Petersen, Kaare Brandt; Pedersen, Michael Syskind. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100302210536/http://www.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf">The Matrix Cookbook</a> (PDF). Archived from <a rel="nofollow" class="external text" href="http://matrixcookbook.com">the original</a> on 2 March 2010. Retrieved 5 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Matrix+Cookbook&rft.aulast=Petersen&rft.aufirst=Kaare+Brandt&rft.au=Pedersen%2C+Michael+Syskind&rft_id=http%3A%2F%2Fmatrixcookbook.com&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> This book uses a mixed layout, i.e. by Y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85bdafd7ed4cebe52fff61221b6adff89228aa8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.821ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}},}"> by X in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28cb5a2232c206504458f88ed3164b93095f6423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.821ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}.}"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuchi" class="citation web cs1">Duchi, John C. <a rel="nofollow" class="external text" href="https://web.stanford.edu/~jduchi/projects/matrix_prop.pdf">"Properties of the Trace and Matrix Derivatives"</a> (PDF). Stanford University. Retrieved 5 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Properties+of+the+Trace+and+Matrix+Derivatives&rft.pub=Stanford+University&rft.aulast=Duchi&rft.aufirst=John+C&rft_id=https%3A%2F%2Fweb.stanford.edu%2F~jduchi%2Fprojects%2Fmatrix_prop.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> See <a href="/wiki/Determinant#Derivative" title="Determinant">Determinant § Derivative</a> for the derivation. </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGiles2008" class="citation conference cs1">Giles, Mike B. (2008). "Collected matrix derivative results for forward and reverse mode algorithmic differentiation". In Bischof, Christian H.; Bücker, H. Martin; Hovland, Paul; Naumann, Uwe; Utke, Jean (eds.). Advances in Automatic Differentiation. Lecture Notes in Computational Science and Engineering. Vol. 64. Berlin: Springer. pp. 35–44. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-68942-3_4">10.1007/978-3-540-68942-3_4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-68935-5" title="Special:BookSources/978-3-540-68935-5"><bdi>978-3-540-68935-5</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2531677">2531677</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Collected+matrix+derivative+results+for+forward+and+reverse+mode+algorithmic+differentiation&rft.btitle=Advances+in+Automatic+Differentiation&rft.place=Berlin&rft.series=Lecture+Notes+in+Computational+Science+and+Engineering&rft.pages=35-44&rft.pub=Springer&rft.date=2008&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2531677%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-540-68942-3_4&rft.isbn=978-3-540-68935-5&rft.aulast=Giles&rft.aufirst=Mike+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a rel="nofollow" class="external text" href="https://www.ias.edu/sites/default/files/sns/files/1-matrixlog_tex(1).pdf">Unpublished memo</a> by S Adler (IAS) </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFangZhang1990" class="citation book cs1"><a href="/wiki/Kai-Tai_Fang" class="mw-redirect" title="Kai-Tai Fang">Fang, Kai-Tai</a>; Zhang, Yao-Ting (1990). Generalized multivariate analysis. Science Press (Beijing) and Springer-Verlag (Berlin). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3540176519" title="Special:BookSources/3540176519"><bdi>3540176519</bdi></a>. 9783540176510.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generalized+multivariate+analysis&rft.pub=Science+Press+%28Beijing%29+and+Springer-Verlag+%28Berlin%29&rft.date=1990&rft.isbn=3540176519&rft.aulast=Fang&rft.aufirst=Kai-Tai&rft.au=Zhang%2C+Yao-Ting&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPanFang2007" class="citation book cs1">Pan, Jianxin; <a href="/wiki/Kaitai_Fang" class="mw-redirect" title="Kaitai Fang">Fang, Kaitai</a> (2007). Growth curve models and statistical diagnostics. Beijing: Science Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387950532" title="Special:BookSources/9780387950532"><bdi>9780387950532</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Growth+curve+models+and+statistical+diagnostics&rft.place=Beijing&rft.pub=Science+Press&rft.date=2007&rft.isbn=9780387950532&rft.aulast=Pan&rft.aufirst=Jianxin&rft.au=Fang%2C+Kaitai&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKollovon_Rosen2005" class="citation book cs1">Kollo, Tõnu; von Rosen, Dietrich (2005). Advanced multivariate statistics with matrices. Dordrecht: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-3418-3" title="Special:BookSources/978-1-4020-3418-3"><bdi>978-1-4020-3418-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+multivariate+statistics+with+matrices&rft.place=Dordrecht&rft.pub=Springer&rft.date=2005&rft.isbn=978-1-4020-3418-3&rft.aulast=Kollo&rft.aufirst=T%C3%B5nu&rft.au=von+Rosen%2C+Dietrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMagnusNeudecker2019" class="citation book cs1">Magnus, Jan; Neudecker, Heinz (2019). Matrix differential calculus with applications in statistics and econometrics. New York: John Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781119541202" title="Special:BookSources/9781119541202"><bdi>9781119541202</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+differential+calculus+with+applications+in+statistics+and+econometrics&rft.place=New+York&rft.pub=John+Wiley&rft.date=2019&rft.isbn=9781119541202&rft.aulast=Magnus&rft.aufirst=Jan&rft.au=Neudecker%2C+Heinz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiuLeivaZhuangMa2022" class="citation journal cs1">Liu, Shuangzhe; Leiva, Victor; Zhuang, Dan; Ma, Tiefeng; Figueroa-Zúñiga, Jorge I. (2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jmva.2021.104849">"Matrix differential calculus with applications in the multivariate linear model and its diagnostics"</a>. Journal of Multivariate Analysis. 188: 104849. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jmva.2021.104849">10.1016/j.jmva.2021.104849</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Multivariate+Analysis&rft.atitle=Matrix+differential+calculus+with+applications+in+the+multivariate+linear+model+and+its+diagnostics&rft.volume=188&rft.pages=104849&rft.date=2022&rft_id=info%3Adoi%2F10.1016%2Fj.jmva.2021.104849&rft.aulast=Liu&rft.aufirst=Shuangzhe&rft.au=Leiva%2C+Victor&rft.au=Zhuang%2C+Dan&rft.au=Ma%2C+Tiefeng&rft.au=Figueroa-Z%C3%BA%C3%B1iga%2C+Jorge+I.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.jmva.2021.104849&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiuTrenklerKollovon_Rosen2023" class="citation journal cs1">Liu, Shuangzhe; Trenkler, Götz; Kollo, Tõnu; von Rosen, Dietrich; Baksalary, Oskar Maria (2023). "Professor Heinz Neudecker and matrix differential calculus". Statistical Papers. 65 (4): 2605–2639. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00362-023-01499-w">10.1007/s00362-023-01499-w</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:263661094">263661094</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Papers&rft.atitle=Professor+Heinz+Neudecker+and+matrix+differential+calculus&rft.volume=65&rft.issue=4&rft.pages=2605-2639&rft.date=2023&rft_id=info%3Adoi%2F10.1007%2Fs00362-023-01499-w&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A263661094%23id-name%3DS2CID&rft.aulast=Liu&rft.aufirst=Shuangzhe&rft.au=Trenkler%2C+G%C3%B6tz&rft.au=Kollo%2C+T%C3%B5nu&rft.au=von+Rosen%2C+Dietrich&rft.au=Baksalary%2C+Oskar+Maria&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=31" title="Edit section: Further reading">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbadirMagnus2005" class="citation book cs1">Abadir, Karim M.; Magnus, Jan R. (2005). Matrix algebra. Econometric Exercises. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-511-64796-3" title="Special:BookSources/978-0-511-64796-3"><bdi>978-0-511-64796-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/569411497">569411497</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+algebra&rft.place=Cambridge&rft.series=Econometric+Exercises&rft.pub=Cambridge+University+Press&rft.date=2005&rft_id=info%3Aoclcnum%2F569411497&rft.isbn=978-0-511-64796-3&rft.aulast=Abadir&rft.aufirst=Karim+M.&rft.au=Magnus%2C+Jan+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLax2007" class="citation book cs1">Lax, Peter D. (2007). "9. Calculus of Vector- and Matrix-Valued Functions". Linear algebra and its applications (2nd ed.). Hoboken, N.J.: Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-75156-4" title="Special:BookSources/978-0-471-75156-4"><bdi>978-0-471-75156-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=9.+Calculus+of+Vector-+and+Matrix-Valued+Functions&rft.btitle=Linear+algebra+and+its+applications&rft.place=Hoboken%2C+N.J.&rft.edition=2nd&rft.pub=Wiley-Interscience&rft.date=2007&rft.isbn=978-0-471-75156-4&rft.aulast=Lax&rft.aufirst=Peter+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMagnus2010" class="citation journal cs1">Magnus, Jan R. (October 2010). "On the concept of matrix derivative". Journal of Multivariate Analysis. 101 (9): 2200–2206. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jmva.2010.05.005">10.1016/j.jmva.2010.05.005</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Multivariate+Analysis&rft.atitle=On+the+concept+of+matrix+derivative&rft.volume=101&rft.issue=9&rft.pages=2200-2206&rft.date=2010-10&rft_id=info%3Adoi%2F10.1016%2Fj.jmva.2010.05.005&rft.aulast=Magnus&rft.aufirst=Jan+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+calculus" class="Z3988">. Note that this Wikipedia article has been nearly completely revised from the version criticized in this article.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=32" title="Edit section: External links">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Software">Software</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=33" title="Edit section: Software">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://www.matrixcalculus.org/">MatrixCalculus.org</a>, a website for evaluating matrix calculus expressions symbolically</li> <li><a rel="nofollow" class="external text" href="https://math.ucsd.edu/~ncalg/">NCAlgebra</a>, an open-source <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> package that has some matrix calculus functionality</li> <li><a href="/wiki/SymPy" title="SymPy">SymPy</a> supports symbolic matrix derivatives in its <a rel="nofollow" class="external text" href="https://docs.sympy.org/latest/modules/matrices/expressions.html">matrix expression module</a>, as well as symbolic tensor derivatives in its <a rel="nofollow" class="external text" href="https://docs.sympy.org/latest/modules/tensor/array_expressions.html">array expression module</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Information">Information</h3>[<a href="/w/index.php?title=Matrix_calculus&action=edit&section=34" title="Edit section: Information">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120630192238/http://www.psi.toronto.edu/matrix/calculus.html">Matrix Reference Manual</a>, Mike Brookes, <a href="/wiki/Imperial_College_London" title="Imperial College London">Imperial College London</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.atmos.washington.edu/~dennis/MatrixCalculus.pdf">Matrix Differentiation (and some other stuff)</a>, Randal J. Barnes, Department of Civil Engineering, University of Minnesota.</li> <li><a rel="nofollow" class="external text" href="http://www4.ncsu.edu/~pfackler/MatCalc.pdf">Notes on Matrix Calculus</a>, Paul L. Fackler, <a href="/wiki/North_Carolina_State_University" title="North Carolina State University">North Carolina State University</a>.</li> <li><a rel="nofollow" class="external text" href="https://wiki.inf.ed.ac.uk/twiki/pub/CSTR/ListenSemester1_2006_7/slide.pdf">Matrix Differential Calculus</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120916044332/https://wiki.inf.ed.ac.uk/twiki/pub/CSTR/ListenSemester1_2006_7/slide.pdf">Archived</a> 2012-09-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> (slide presentation), Zhang Le, <a href="/wiki/University_of_Edinburgh" title="University of Edinburgh">University of Edinburgh</a>.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120526142207/http://www.econ.ku.dk/metrics/Econometrics2_05_II/LectureNotes/matrixdiff.pdf">Introduction to Vector and Matrix Differentiation</a> (notes on matrix differentiation, in the context of <a href="/wiki/Econometrics" title="Econometrics">Econometrics</a>), Heino Bohn Nielsen.</li> <li><a rel="nofollow" class="external text" href="http://mpra.ub.uni-muenchen.de/1239/1/MPRA_paper_1239.pdf">A note on differentiating matrices</a> (notes on matrix differentiation), Pawel Koval, from Munich Personal RePEc Archive.</li> <li><a rel="nofollow" class="external text" href="http://www.personal.rdg.ac.uk/~sis01xh/teaching/CY4C9/ANN3.pdf">Vector/Matrix Calculus</a> More notes on matrix differentiation.</li> <li><a rel="nofollow" class="external text" href="http://www.cs.nyu.edu/~roweis/notes/matrixid.pdf">Matrix Identities</a> (notes on matrix differentiation), Sam Roweis.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Calculus" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Calculus_topics" title="Template:Calculus topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus_topics" title="Template talk:Calculus topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus_topics" title="Special:EditPage/Template:Calculus topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Calculus" style="font-size:114%;margin:0 4em"><a href="/wiki/Calculus" title="Calculus">Calculus</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_theorem" title="Binomial theorem">Binomial theorem</a></li> <li><a href="/wiki/Concave_function" title="Concave function">Concave function</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous function</a></li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Finite difference</a></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li> <li><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a></li> <li><a href="/wiki/Linear_function" title="Linear function">Linear function</a></li> <li><a href="/wiki/Radian" title="Radian">Radian</a></li> <li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Secant_line" title="Secant line">Secant</a></li> <li><a href="/wiki/Slope" title="Slope">Slope</a></li> <li><a href="/wiki/Tangent" title="Tangent">Tangent</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li> <li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limit of a function</a> <ul><li><a href="/wiki/One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li> <li><a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a></li> <li><a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a> <ul><li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><a href="/wiki/Newton%27s_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules of differentiation</a> <ul><li><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a> <ul><li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li> <li>Other techniques <ul><li><a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li> <li><a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a class="mw-selflink selflink">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></li> <li><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a href="/wiki/Regiomontanus%27_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li> <li><a href="/wiki/Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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Matrix calculus - Wikipedia