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class="mw-page-title-main">Multilinear algebra</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81lxebra_multilineal" title="Álxebra multilineal – Asturian" lang="ast" hreflang="ast" data-title="Álxebra multilineal" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BF_%D2%BB%D1%8B%D2%99%D1%8B%D2%A1%D0%BB%D1%8B_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Күп һыҙыҡлы алгебра – Bashkir" lang="ba" hreflang="ba" data-title="Күп һыҙыҡлы алгебра" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Полилинейна алгебра – Bulgarian" lang="bg" hreflang="bg" data-title="Полилинейна алгебра" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_multilineal" title="Àlgebra multilineal – Catalan" lang="ca" hreflang="ca" data-title="Àlgebra multilineal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BB%D0%B8%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Полилинилле алгебра – Chuvash" lang="cv" hreflang="cv" data-title="Полилинилле алгебра" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Multilineare_Algebra" title="Multilineare Algebra – German" lang="de" hreflang="de" data-title="Multilineare Algebra" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BF%CE%BB%CF%85%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%B9%CE%BA%CE%AE_%CE%AC%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1" title="Πολυγραμμική άλγεβρα – Greek" lang="el" hreflang="el" data-title="Πολυγραμμική άλγεβρα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_multilineal" title="Álgebra multilineal – Spanish" lang="es" hreflang="es" data-title="Álgebra multilineal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aljebra_multilineal" title="Aljebra multilineal – Basque" lang="eu" hreflang="eu" data-title="Aljebra multilineal" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_multilin%C3%A9aire" title="Algèbre multilinéaire – French" lang="fr" hreflang="fr" data-title="Algèbre multilinéaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81lxebra_multilinear" title="Álxebra multilinear – Galician" lang="gl" hreflang="gl" data-title="Álxebra multilinear" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%EC%A4%91%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="다중선형대수학 – Korean" lang="ko" hreflang="ko" data-title="다중선형대수학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aljabar_multilinear" title="Aljabar multilinear – Indonesian" lang="id" hreflang="id" data-title="Aljabar multilinear" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%9A%E9%87%8D%E7%B7%9A%E5%9E%8B%E4%BB%A3%E6%95%B0" title="多重線型代数 – Japanese" lang="ja" hreflang="ja" data-title="多重線型代数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algebra_wieloliniowa" title="Algebra wieloliniowa – Polish" lang="pl" hreflang="pl" data-title="Algebra wieloliniowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_multilinear" title="Álgebra multilinear – Portuguese" lang="pt" hreflang="pt" data-title="Álgebra multilinear" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Полилинейная алгебра – Russian" lang="ru" hreflang="ru" data-title="Полилинейная алгебра" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Polilinearna_algebra" title="Polilinearna algebra – Serbian" lang="sr" hreflang="sr" data-title="Polilinearna algebra" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Alhebrang_multilinyar" title="Alhebrang multilinyar – Tagalog" lang="tl" hreflang="tl" data-title="Alhebrang multilinyar" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87okludo%C4%9Frusal_cebir" title="Çokludoğrusal cebir – Turkish" lang="tr" hreflang="tr" data-title="Çokludoğrusal cebir" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D0%B0%D0%B3%D0%B0%D1%82%D0%BE%D0%BB%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Багатолінійна алгебра – Ukrainian" lang="uk" hreflang="uk" data-title="Багатолінійна алгебра" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1i_s%E1%BB%91_%C4%91a_tuy%E1%BA%BFn_t%C3%ADnh" title="Đại số đa tuyến tính – Vietnamese" lang="vi" hreflang="vi" data-title="Đại số đa tuyến tính" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%9A%E9%87%8D%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" title="多重線性代數 – Cantonese" lang="yue" hreflang="yue" data-title="多重線性代數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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with the functions being <a href="/wiki/Linear_map" title="Linear map">linear maps</a> with respect to each argument. It involves concepts such as <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, <a href="/wiki/Tensor" title="Tensor">tensors</a>, <a href="/wiki/Multivector" title="Multivector">multivectors</a>, <a href="/wiki/System_of_linear_equations" title="System of linear equations">systems of linear equations</a>, <a href="/wiki/Higher-dimensional_space" class="mw-redirect" title="Higher-dimensional space">higher-dimensional spaces</a>, <a href="/wiki/Determinant" title="Determinant">determinants</a>, <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner</a> and <a href="/wiki/Outer_product" title="Outer product">outer</a> products, and <a href="/wiki/Dual_space" title="Dual space">dual spaces</a>. It is a mathematical tool used in <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, <a href="/wiki/Physics" title="Physics">physics</a>, and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Origin">Origin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multilinear_algebra&action=edit&section=1" title="Edit section: Origin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While many theoretical concepts and applications involve <a href="/wiki/Vector_space" title="Vector space">single vectors</a>, mathematicians such as <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a> considered structures involving pairs, triplets, and <a href="/wiki/Multivector" title="Multivector">multivectors</a> that generalize <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a>. With multiple combinational possibilities, the space of <a href="/wiki/Multivector" title="Multivector">multivectors</a> expands to 2<sup><i>n</i></sup> dimensions, where <i>n</i> is the dimension of the relevant vector space.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Determinant#Abstract_formulation" title="Determinant">determinant can be formulated abstractly</a> using the structures of multilinear algebra. </p><p>Multilinear algebra appears in the study of the mechanical response of materials to stress and strain, involving various moduli of <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elasticity</a>. The term "<a href="/wiki/Tensor" title="Tensor">tensor</a>" describes elements within the multilinear space due to its added structure. Despite Grassmann's early work in 1844 with his <i><a href="/wiki/Ausdehnungslehre" class="mw-redirect" title="Ausdehnungslehre">Ausdehnungslehre</a></i>, which was also republished in 1862, the subject was initially not widely understood, as even ordinary linear algebra posed many challenges at the time. </p><p>The concepts of multilinear algebra find applications in certain studies of <a href="/wiki/Multivariate_calculus" class="mw-redirect" title="Multivariate calculus">multivariate calculus</a> and <a href="/wiki/Manifold" title="Manifold">manifolds</a>, particularly concerning the <a href="/wiki/Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a>. <a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">Infinitesimal differentials</a> encountered in single-variable calculus are transformed into <a href="/wiki/Differential_forms" class="mw-redirect" title="Differential forms">differential forms</a> in <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariate calculus</a>, and their manipulation is carried out using <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Following Grassmann, developments in multilinear algebra were made by <a href="/wiki/Victor_Schlegel" title="Victor Schlegel">Victor Schlegel</a> in 1872 with the publication of the first part of his <i>System der Raumlehre</i><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and by <a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a>. Notably, significant advancements came through the work of <a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a> and <a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> particularly in the form of <i>absolute differential calculus</i> within multilinear algebra. <a href="/wiki/Marcel_Grossmann" title="Marcel Grossmann">Marcel Grossmann</a> and <a href="/wiki/Michele_Besso" title="Michele Besso">Michele Besso</a> introduced this form to <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, and in 1915, Einstein's publication on <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, explaining the <a href="/wiki/Precession_of_the_perihelion_of_Mercury" class="mw-redirect" title="Precession of the perihelion of Mercury">precession of Mercury's perihelion</a>, established multilinear algebra and tensors as important mathematical tools in physics. </p><p>In 1958, <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a> included a chapter on multilinear algebra titled "<i>Algèbre Multilinéaire</i>" in his series <a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Éléments de mathématique</a>, specifically within the algebra book. The chapter covers topics such as bilinear functions, the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of two <a href="/wiki/Module_(ring_theory)" class="mw-redirect" title="Module (ring theory)">modules</a>, and the properties of tensor products.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multilinear_algebra&action=edit&section=2" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Multilinear algebra concepts find applications in various areas, including: </p> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Classical_treatment_of_tensors" class="mw-redirect" title="Classical treatment of tensors">Classical treatment of tensors</a></li> <li><a href="/wiki/Dyadic_tensor" class="mw-redirect" title="Dyadic tensor">Dyadic tensor</a></li> <li><a href="/wiki/Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Multilinear_subspace_learning" title="Multilinear subspace learning">Multilinear subspace learning</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multilinear_algebra&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li> <li><a href="/wiki/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed and exact differential forms</a></li> <li><a href="/wiki/Component-free_treatment_of_tensors" class="mw-redirect" title="Component-free treatment of tensors">Component-free treatment of tensors</a></li> <li><a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual space</a></li> <li><a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">Inner product</a></li> <li><a href="/wiki/Outer_product" title="Outer product">Outer product</a></li> <li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Multilinear_form" title="Multilinear form">Multilinear form</a></li> <li><a href="/wiki/Pseudoscalar_(mathematics)" class="mw-redirect" title="Pseudoscalar (mathematics)">Pseudoscalar</a></li> <li><a href="/wiki/Pseudovector" title="Pseudovector">Pseudovector</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a>, <a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Symmetric_algebra" title="Symmetric algebra">Symmetric algebra</a>, <a href="/wiki/Symmetric_power" title="Symmetric power">Symmetric power</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multilinear_algebra&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFPandeyVenugopalLeib2024" class="citation journal cs1">Pandey, Divyanshu; Venugopal, Adithya; Leib, Harry (2024). <a rel="nofollow" class="external text" href="https://doi.org/10.3389%2Ffams.2023.1259836">"Linear to multi-linear algebra and systems using tensors"</a>. <i>Frontiers in Applied Mathematics and Statistics</i>. <b>9</b>. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2304.10658">2304.10658</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3389%2Ffams.2023.1259836">10.3389/fams.2023.1259836</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2297-4687">2297-4687</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Frontiers+in+Applied+Mathematics+and+Statistics&rft.atitle=Linear+to+multi-linear+algebra+and+systems+using+tensors&rft.volume=9&rft.date=2024&rft_id=info%3Aarxiv%2F2304.10658&rft.issn=2297-4687&rft_id=info%3Adoi%2F10.3389%2Ffams.2023.1259836&rft.aulast=Pandey&rft.aufirst=Divyanshu&rft.au=Venugopal%2C+Adithya&rft.au=Leib%2C+Harry&rft_id=https%3A%2F%2Fdoi.org%2F10.3389%252Ffams.2023.1259836&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultilinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrassmann2000" class="citation book cs1">Grassmann, Hermann (2000) [1862]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yeGPeaPVLKoC&pg=PP1"><i>Extension Theory</i></a> [<i>Die Ausdehnungslehre</i>]. Translated by Kannenberg, Lloyd. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-9049-3" title="Special:BookSources/978-0-8218-9049-3"><bdi>978-0-8218-9049-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Extension+Theory&rft.pub=American+Mathematical+Society&rft.date=2000&rft.isbn=978-0-8218-9049-3&rft.aulast=Grassmann&rft.aufirst=Hermann&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyeGPeaPVLKoC%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultilinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFleming1977" class="citation book cs1">Fleming, Wendell H. (1977). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-1-4684-9461-7_7">"Exterior algebra and differential calculus"</a>. <i>Functions of Several Variables</i>. Undergraduate Texts in Mathematics (2nd ed.). Springer. pp. 275–320. <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%2F978-1-4684-9461-7_7">10.1007/978-1-4684-9461-7_7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4684-9461-7" title="Special:BookSources/978-1-4684-9461-7"><bdi>978-1-4684-9461-7</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/2401829">2401829</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Exterior+algebra+and+differential+calculus&rft.btitle=Functions+of+Several+Variables&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=275-320&rft.edition=2nd&rft.pub=Springer&rft.date=1977&rft_id=info%3Aoclcnum%2F2401829&rft_id=info%3Adoi%2F10.1007%2F978-1-4684-9461-7_7&rft.isbn=978-1-4684-9461-7&rft.aulast=Fleming&rft.aufirst=Wendell+H.&rft_id=https%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F978-1-4684-9461-7_7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultilinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchlegel2018" class="citation book cs1">Schlegel, Victor (2018). <i>System der Raumlehre: Nach den Prinzipien der Grassmann'schen Ausdehnungslehre und als Einleitung in Dieselbe; Geometrie; Die Gebiete des Punktes, der Geraden, der Ebene</i>. Forgotten Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-364-22177-8" title="Special:BookSources/978-0-364-22177-8"><bdi>978-0-364-22177-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=System+der+Raumlehre%3A+Nach+den+Prinzipien+der+Grassmann%27schen+Ausdehnungslehre+und+als+Einleitung+in+Dieselbe%3B+Geometrie%3B+Die+Gebiete+des+Punktes%2C+der+Geraden%2C+der+Ebene&rft.pub=Forgotten+Books&rft.date=2018&rft.isbn=978-0-364-22177-8&rft.aulast=Schlegel&rft.aufirst=Victor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultilinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRicci-CurbastroLevi-Civita1900" class="citation journal cs1"><a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Ricci-Curbastro, Gregorio</a>; <a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Levi-Civita, Tullio</a> (1900). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1428270">"Méthodes de calcul différentiel absolu et leurs applications"</a>. <i><a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>. <b>54</b> (1): 125–201. <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%2FBF01454201">10.1007/BF01454201</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1432-1807">1432-1807</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:120009332">120009332</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=M%C3%A9thodes+de+calcul+diff%C3%A9rentiel+absolu+et+leurs+applications&rft.volume=54&rft.issue=1&rft.pages=125-201&rft.date=1900&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120009332%23id-name%3DS2CID&rft.issn=1432-1807&rft_id=info%3Adoi%2F10.1007%2FBF01454201&rft.aulast=Ricci-Curbastro&rft.aufirst=Gregorio&rft.au=Levi-Civita%2C+Tullio&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1428270&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultilinear+algebra" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a> (1958) <i>Algèbra Multilinéair</i>, chapter 3 of book 2 <i>Algebra</i>, in <a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Éléments de mathématique</a>, Paris: Hermann</span> </li> </ol></div></div> <ul><li>Greub, W. H. (1967) <i>Multilinear Algebra</i>, Springer</li> <li><a href="/wiki/Douglas_Northcott" title="Douglas Northcott">Douglas Northcott</a> (1984) <i>Multilinear Algebra</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-26269-0" title="Special:BookSources/0-521-26269-0">0-521-26269-0</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShaw1983" class="citation book cs1">Shaw, Ronald (1983). <i>Multilinear algebra and group representations</i>. Vol. 2. <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-639202-9" title="Special:BookSources/978-0-12-639202-9"><bdi>978-0-12-639202-9</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/59106339">59106339</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multilinear+algebra+and+group+representations&rft.pub=Academic+Press&rft.date=1983&rft_id=info%3Aoclcnum%2F59106339&rft.isbn=978-0-12-639202-9&rft.aulast=Shaw&rft.aufirst=Ronald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultilinear+algebra" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist 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aria-labelledby="Linear_algebra" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Linear_algebra" title="Template:Linear algebra"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Linear_algebra" title="Template talk:Linear algebra"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Linear_algebra" title="Special:EditPage/Template:Linear algebra"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Linear_algebra" 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projection">Vector projection</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Linear projection</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Linear_combination" title="Linear combination">Linear combination</a></li> <li><a href="/wiki/Multilinear_map" title="Multilinear map">Multilinear map</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Change_of_basis" title="Change of basis">Change of basis</a></li> <li><a href="/wiki/Row_and_column_vectors" title="Row and column vectors">Row and column vectors</a></li> <li><a href="/wiki/Row_and_column_spaces" title="Row and column spaces">Row and column spaces</a></li> <li><a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">Kernel</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a></li> <li><a href="/wiki/System_of_linear_equations" title="System of linear equations">Linear equations</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Euclidean_space" title="Euclidean space"><img alt="Three dimensional Euclidean space" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, 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space (linear algebra)">Quotient</a></li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Subspace</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical</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/Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a></li> <li><a href="/wiki/Numerical_stability" title="Numerical stability">Numerical stability</a></li> <li><a href="/wiki/Basic_Linear_Algebra_Subprograms" title="Basic Linear Algebra Subprograms">Basic Linear Algebra Subprograms</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li> <li><a href="/wiki/Comparison_of_linear_algebra_libraries" title="Comparison of linear algebra libraries">Comparison of linear algebra libraries</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Tensors" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed 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:Tensors" title="Template:Tensors"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tensors" title="Template talk:Tensors"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tensors" title="Special:EditPage/Template:Tensors"><abbr title="Edit this template">e</abbr></a></li></ul></div><div 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geometry">Differential geometry</a></li> <li><a href="/wiki/Dyadics" title="Dyadics">Dyadic algebra</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li> <li><a class="mw-selflink selflink">Multilinear algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="/wiki/Physics" title="Physics">Physics</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></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/Computer_vision" title="Computer vision">Computer vision</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li> <li><a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a></li> <li><a href="/wiki/Index_notation" title="Index notation">Index notation</a></li> <li><a href="/wiki/Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li> <li><a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li> <li><a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li> <li><a href="/wiki/Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li> <li><a href="/wiki/Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li> <li><a href="/wiki/Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor<br />definitions</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/Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a></li> <li><a href="/wiki/Tensor_density" title="Tensor density">Tensor density</a></li> <li><a href="/wiki/Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li> <li><a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Tensor_operator" title="Tensor operator">Tensor operator</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li> <li><a href="/wiki/Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />abstractions</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/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</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/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li> <li><a href="/wiki/Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li> <li><a href="/wiki/Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li> <li><a href="/wiki/Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li> <li><a 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