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matrix in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="linear_algebra">Linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/higher+linear+algebra">higher linear algebra</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-module">(∞,n)-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field">field</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-field">∞-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/rational+vector+space">rational vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+basis">orthogonal basis</a>, <a class="existingWikiWord" href="/nlab/show/orthonormal+basis">orthonormal basis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a>, <a class="existingWikiWord" href="/nlab/show/antilinear+map">antilinear map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> (<a class="existingWikiWord" href="/nlab/show/square+matrix">square</a>, <a class="existingWikiWord" href="/nlab/show/invertible+matrix">invertible</a>, <a class="existingWikiWord" href="/nlab/show/diagonal+matrix">diagonal</a>, <a class="existingWikiWord" href="/nlab/show/hermitian+matrix">hermitian</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+matrix">symmetric</a>, …)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>, <a class="existingWikiWord" href="/nlab/show/matrix+group">matrix group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eigenspace">eigenspace</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a>, <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></p> <p><a class="existingWikiWord" href="/nlab/show/Gram-Schmidt+process">Gram-Schmidt process</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <p>(…)</p> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>For <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> and a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n\times m</annotation></semantics></math> <strong>matrix</strong> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">M:[n]\times[m]\rightarrow X</annotation></semantics></math> from the Cartesian product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>×</mo><mo stretchy="false">[</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[n]\times[m]</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Often one uses the term in a context where one can add and multiply matrices using <a class="existingWikiWord" href="/nlab/show/matrix+calculus">matrix calculus</a>. Addition of matrices of the same dimension requires <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to have an “addition” operation; multiplication of matrices requires <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to also have a “multiplication” operation. Usually <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is at least a <a class="existingWikiWord" href="/nlab/show/rig">rig</a> and often a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> or a <a class="existingWikiWord" href="/nlab/show/field">field</a>.</p> <p>More generally, for arbitrary sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> we can define an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\times B</annotation></semantics></math>-matrix to be a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A\times B\to X</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has some kind of “infinitary sums” as well as finite “products”, then we can also multiply matrices of this sort: e.g. if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the set of objects of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with arbitrary <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>.</p> <p>Note that if the structure on X is such that matrix multiplication is associative and there exist identity matrices, then matrices over X may be taken as the morphisms of a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mat</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">Mat_X</annotation></semantics></math> whose composition is matrix multiplication. This is in particular the case if X is a field, and so many basic theorems of linear algebra may be understood as concerning functors from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Vect</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">Vect_X</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mat</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">Mat_X</annotation></semantics></math> and natural transformations between such functors.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+multiplication">matrix multiplication</a>, <a class="existingWikiWord" href="/nlab/show/matrix+calculus">matrix calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+decomposition">matrix decomposition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/square+matrix">square matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diagonal+matrix">diagonal matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+matrix">elementary matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/block+matrix">block matrix</a> (<a class="existingWikiWord" href="/nlab/show/partitioned+matrix">partitioned matrix</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/row+echelon+matrix">row echelon matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rank+of+a+matrix">rank of a matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverse+matrix">inverse matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+equivalence">matrix equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monomial+matrix">monomial matrix</a>, <a class="existingWikiWord" href="/nlab/show/permutation+matrix">permutation matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+submatrix">principal submatrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+group">matrix group</a>, <a class="existingWikiWord" href="/nlab/show/matrix+Lie+group">matrix Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+matrix">unitary matrix</a>. <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangular+matrix">triangular matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Smith+normal+form">Smith normal form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stochastic+matrix">stochastic matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjacency+matrix">adjacency matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>, <a class="existingWikiWord" href="/nlab/show/special+linear+group">special linear group</a>, <a class="existingWikiWord" href="/nlab/show/matrix+mechanics">matrix mechanics</a>, <a class="existingWikiWord" href="/nlab/show/matrix+theory">matrix theory</a>, <a class="existingWikiWord" href="/nlab/show/matrix+Hopf+algebra">matrix Hopf algebra</a>, <a class="existingWikiWord" href="/nlab/show/matrix+Lie+algebra">matrix Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/matrix+Lie+group">matrix Lie group</a>, <a class="existingWikiWord" href="/nlab/show/classical+Lie+group">classical Lie group</a>, <a class="existingWikiWord" href="/nlab/show/universal+localization">universal localization</a>, <a class="existingWikiWord" href="/nlab/show/tensor+calculus">tensor calculus</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/characteristic+polynomial">characteristic polynomial</a> (Cayley-Hamilton theorem), <a class="existingWikiWord" href="/nlab/show/spectral+curve">spectral curve</a></p> </li> </ul> <p>Special cases: <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/classical+r-matrix">classical r-matrix</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a>, <a class="existingWikiWord" href="/nlab/show/hermitian+matrix">hermitian matrix</a>, <a class="existingWikiWord" href="/nlab/show/skew-symmetric+matrix">skew-symmetric matrix</a>, <a class="existingWikiWord" href="/nlab/show/quantum+Yang-Baxter+matrix">quantum Yang-Baxter matrix</a>, <a class="existingWikiWord" href="/nlab/show/random+matrix">random matrix</a>, <a class="existingWikiWord" href="/nlab/show/skew-symmetric+matrix">skew-symmetric matrix</a></p> <p>Operations on/with matrices: <a class="existingWikiWord" href="/nlab/show/transpose+matrix">transpose matrix</a>, <a class="existingWikiWord" href="/nlab/show/adjoint+matrix">adjoint matrix</a> <a class="existingWikiWord" href="/nlab/show/trace">trace</a>, <a class="existingWikiWord" href="/nlab/show/matrix+factorization">matrix factorization</a>, <a class="existingWikiWord" href="/nlab/show/Gauss+decomposition">Gauss decomposition</a>, <a class="existingWikiWord" href="/nlab/show/Gram-Schmidt+process">Gram-Schmidt process</a></p> <p>Determinants and <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a> like notions, and special cases: <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a>, <a class="existingWikiWord" href="/nlab/show/Berezinian">Berezinian</a>,<a class="existingWikiWord" href="/nlab/show/Jacobian">Jacobian</a>, <a class="existingWikiWord" href="/nlab/show/Pfaffian">Pfaffian</a>, <a class="existingWikiWord" href="/nlab/show/hafnian">hafnian</a>, <a class="existingWikiWord" href="/nlab/show/Wronskian">Wronskian</a>, <a class="existingWikiWord" href="/nlab/show/resultant">resultant</a>, <a class="existingWikiWord" href="/nlab/show/discriminant">discriminant</a></p> <h2 id="references">References</h2> <p>Historical origins:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Arthur+Cayley">Arthur Cayley</a>: <em>A Memoir on the Theory of Matrices</em>, Philosophical Transactions of the Royal Society of London, <strong>148</strong> (1858) 17-37 [<a href="https://www.jstor.org/stable/108649">jstor:108649</a>]</li> </ul> <p>Textbook accounts:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Igor+R.+Shafarevich">Igor R. Shafarevich</a>, <a class="existingWikiWord" href="/nlab/show/Alexey+O.+Remizov">Alexey O. Remizov</a>: §2 in: <em>Linear Algebra and Geometry</em> (2012) [<a href="https://doi.org/10.1007/978-3-642-30994-6">doi:10.1007/978-3-642-30994-6</a>, <a href="https://maa.org/press/maa-reviews/linear-algebra-and-geometry">MAA-review</a>]</li> </ul> <p>See also:</p> <ul> <li>Wikipedia: <a href="https://en.wikipedia.org/wiki/Matrix_(mathematics)">Matrix (mathematics)</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 9, 2024 at 10:55:27. 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