Matrix (mathematics)

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class="mf-section-0" id="mf-section-0"> <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">"Matrix theory" redirects here. For the physics topic, see <a href="/wiki/Matrix_theory_(physics)" title="Matrix theory (physics)">Matrix theory (physics)</a>. For other uses of "Matrix", see <a href="/wiki/Matrix_(disambiguation)" class="mw-redirect mw-disambig" title="Matrix (disambiguation)">Matrix (disambiguation)</a>.</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a matrix (<abbr title="plural form">pl.</abbr>: matrices) is a <a href="/wiki/Rectangle" title="Rectangle">rectangular</a> array or table of <a href="/wiki/Number" title="Number">numbers</a>, <a href="/wiki/Symbol_(formal)" title="Symbol (formal)">symbols</a>, or <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expressions</a>, with elements or entries arranged in rows and columns, which is used to represent a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a> or property of such an object. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:MatrixLabelled.svg" class="mw-file-description"><img alt="Two tall square brackets with m-many rows each containing n-many subscripted letter 'a' variables. Each letter 'a' is given a row number and column number as its subscript." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/MatrixLabelled.svg/220px-MatrixLabelled.svg.png" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/MatrixLabelled.svg/330px-MatrixLabelled.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/MatrixLabelled.svg/440px-MatrixLabelled.svg.png 2x" data-file-width="140" data-file-height="98"></a><figcaption>An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two <a href="/wiki/Index_notation" title="Index notation">subscripts</a>. For example, a2,1 represents the element at the second row and first column of the matrix.</figcaption></figure> For example, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mo>−</mo> <mn>13</mn> </mtd> </mtr> <mtr> <mtd> <mn>20</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f786996bcfb75972dd77712c90122bc8765269" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.472ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}}"> is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b74d5a424cfb56b99e1060910dbfed284314da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 3}"> matrix", or a matrix of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b74d5a424cfb56b99e1060910dbfed284314da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 3}">. Matrices are commonly related to <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>. Notable exceptions include <a href="/wiki/Incidence_matrix" title="Incidence matrix">incidence matrices</a> and <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrices</a> in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>.<a href="#cite_note-1">[1]</a> This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent <a href="/wiki/Linear_map" title="Linear map">linear maps</a> or may be viewed as such. <a href="/wiki/Square_matrices" class="mw-redirect" title="Square matrices">Square matrices</a>, matrices with the same number of rows and columns, play a major role in matrix theory. Square matrices of a given dimension form a <a href="/wiki/Noncommutative_ring" title="Noncommutative ring">noncommutative ring</a>, which is one of the most common examples of a noncommutative ring. The <a href="/wiki/Determinant" title="Determinant">determinant</a> of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> if and only if it has a nonzero determinant and the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of a square matrix are the roots of a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> determinant. In <a href="/wiki/Geometry" title="Geometry">geometry</a>, matrices are widely used for specifying and representing <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformations</a> (for example <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a>) and <a href="/wiki/Coordinate_change" class="mw-redirect" title="Coordinate change">coordinate changes</a>. In <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly, or through their use in geometry and numerical analysis. Matrix theory is the <a href="/wiki/Branch_of_mathematics" class="mw-redirect" title="Branch of mathematics">branch of mathematics</a> that focuses on the study of matrices. It was initially a sub-branch of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, but soon grew to include subjects related to <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, <a href="/wiki/Algebra" title="Algebra">algebra</a>, <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Definition">1 Definition</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Size">1.1 Size</a></li> </ul> </li> <li class="toclevel-1 tocsection-3"><a href="#Notation">2 Notation</a></li> <li class="toclevel-1 tocsection-4"><a href="#Basic_operations">3 Basic operations</a> <ul> <li class="toclevel-2 tocsection-5"><a href="#Addition,_scalar_multiplication,_subtraction_and_transposition">3.1 Addition, scalar multiplication, subtraction and transposition</a></li> <li class="toclevel-2 tocsection-6"><a href="#Matrix_multiplication">3.2 Matrix multiplication</a></li> <li class="toclevel-2 tocsection-7"><a href="#Row_operations">3.3 Row operations</a></li> <li class="toclevel-2 tocsection-8"><a href="#Submatrix">3.4 Submatrix</a></li> </ul> </li> <li class="toclevel-1 tocsection-9"><a href="#Linear_equations">4 Linear equations</a></li> <li class="toclevel-1 tocsection-10"><a href="#Linear_transformations">5 Linear transformations</a></li> <li class="toclevel-1 tocsection-11"><a href="#Square_matrix">6 Square matrix</a> <ul> <li class="toclevel-2 tocsection-12"><a href="#Main_types">6.1 Main types</a> <ul> <li class="toclevel-3 tocsection-13"><a href="#Diagonal_and_triangular_matrix">6.1.1 Diagonal and triangular matrix</a></li> <li class="toclevel-3 tocsection-14"><a href="#Identity_matrix">6.1.2 Identity matrix</a></li> <li class="toclevel-3 tocsection-15"><a href="#Symmetric_or_skew-symmetric_matrix">6.1.3 Symmetric or skew-symmetric matrix</a></li> <li class="toclevel-3 tocsection-16"><a href="#Invertible_matrix_and_its_inverse">6.1.4 Invertible matrix and its inverse</a></li> <li class="toclevel-3 tocsection-17"><a href="#Definite_matrix">6.1.5 Definite matrix</a></li> <li class="toclevel-3 tocsection-18"><a href="#Orthogonal_matrix">6.1.6 Orthogonal matrix</a></li> </ul> </li> <li class="toclevel-2 tocsection-19"><a href="#Main_operations">6.2 Main operations</a> <ul> <li class="toclevel-3 tocsection-20"><a href="#Trace">6.2.1 Trace</a></li> <li class="toclevel-3 tocsection-21"><a href="#Determinant">6.2.2 Determinant</a></li> <li class="toclevel-3 tocsection-22"><a href="#Eigenvalues_and_eigenvectors">6.2.3 Eigenvalues and eigenvectors</a></li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-23"><a href="#Computational_aspects">7 Computational aspects</a></li> <li class="toclevel-1 tocsection-24"><a href="#Decomposition">8 Decomposition</a></li> <li class="toclevel-1 tocsection-25"><a href="#Abstract_algebraic_aspects_and_generalizations">9 Abstract algebraic aspects and generalizations</a> <ul> <li class="toclevel-2 tocsection-26"><a href="#Matrices_with_more_general_entries">9.1 Matrices with more general entries</a></li> <li class="toclevel-2 tocsection-27"><a href="#Relationship_to_linear_maps">9.2 Relationship to linear maps</a></li> <li class="toclevel-2 tocsection-28"><a href="#Matrix_groups">9.3 Matrix groups</a></li> <li class="toclevel-2 tocsection-29"><a href="#Infinite_matrices">9.4 Infinite matrices</a></li> <li class="toclevel-2 tocsection-30"><a href="#Empty_matrix">9.5 Empty matrix</a></li> </ul> </li> <li class="toclevel-1 tocsection-31"><a href="#Applications">10 Applications</a> <ul> <li class="toclevel-2 tocsection-32"><a href="#Graph_theory">10.1 Graph theory</a></li> <li class="toclevel-2 tocsection-33"><a href="#Analysis_and_geometry">10.2 Analysis and geometry</a></li> <li class="toclevel-2 tocsection-34"><a href="#Probability_theory_and_statistics">10.3 Probability theory and statistics</a></li> <li class="toclevel-2 tocsection-35"><a href="#Symmetries_and_transformations_in_physics">10.4 Symmetries and transformations in physics</a></li> <li class="toclevel-2 tocsection-36"><a href="#Linear_combinations_of_quantum_states">10.5 Linear combinations of quantum states</a></li> <li class="toclevel-2 tocsection-37"><a href="#Normal_modes">10.6 Normal modes</a></li> <li class="toclevel-2 tocsection-38"><a href="#Geometrical_optics">10.7 Geometrical optics</a></li> <li class="toclevel-2 tocsection-39"><a href="#Electronics">10.8 Electronics</a></li> </ul> </li> <li class="toclevel-1 tocsection-40"><a href="#History">11 History</a> <ul> <li class="toclevel-2 tocsection-41"><a href="#Other_historical_usages_of_the_word_%22matrix%22_in_mathematics">11.1 Other historical usages of the word "matrix" in mathematics</a></li> </ul> </li> <li class="toclevel-1 tocsection-42"><a href="#See_also">12 See also</a></li> <li class="toclevel-1 tocsection-43"><a href="#Notes">13 Notes</a></li> <li class="toclevel-1 tocsection-44"><a href="#References">14 References</a> <ul> <li class="toclevel-2 tocsection-45"><a href="#Physics_references">14.1 Physics references</a></li> <li class="toclevel-2 tocsection-46"><a href="#Historical_references">14.2 Historical references</a></li> </ul> </li> <li class="toclevel-1 tocsection-47"><a href="#Further_reading">15 Further reading</a></li> <li class="toclevel-1 tocsection-48"><a href="#External_links">16 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Definition">Definition</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=1" title="Edit section: Definition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> A matrix is a rectangular array of <a href="/wiki/Number" title="Number">numbers</a> (or other mathematical objects), called the entries of the matrix. Matrices are subject to standard <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> such as <a href="#Basic_operations">addition</a> and <a href="#Matrix_multiplication">multiplication</a>.<a href="#cite_note-2">[2]</a> Most commonly, a matrix over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> F is a rectangular array of <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> of F.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> A real matrix and a complex matrix are matrices whose entries are respectively <a href="/wiki/Real_number" title="Real number">real numbers</a> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. More general types of entries are discussed <a href="#Matrices_with_more_general_entries">below</a>. For instance, this is a real matrix: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{bmatrix}-1.3&0.6\\20.4&5.5\\9.7&-6.2\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1.3</mn> </mtd> <mtd> <mn>0.6</mn> </mtd> </mtr> <mtr> <mtd> <mn>20.4</mn> </mtd> <mtd> <mn>5.5</mn> </mtd> </mtr> <mtr> <mtd> <mn>9.7</mn> </mtd> <mtd> <mo>−</mo> <mn>6.2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}-1.3&0.6\\20.4&5.5\\9.7&-6.2\end{bmatrix}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93197553f2e35e42d461b20ee549ea2fde617e1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.499ex; height:9.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}-1.3&0.6\\20.4&5.5\\9.7&-6.2\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 21.499ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93197553f2e35e42d461b20ee549ea2fde617e1e" data-alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}-1.3&0.6\\20.4&5.5\\9.7&-6.2\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> The numbers, symbols, or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively. <div class="mw-heading mw-heading3"><h3 id="Size">Size</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=2" title="Edit section: Size" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The size of a matrix is defined by the number of rows and columns it contains. There is no limit to the number of rows and columns, that a matrix (in the usual sense) can have as long as they are positive integers. A matrix with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {m}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989d8ed6300d470470571f438bbe51f0693fb7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle {m}}"></noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989d8ed6300d470470571f438bbe51f0693fb7b2" data-alt="{\displaystyle {m}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  rows and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b6881107a598c20a60e72fe82bc41a4a1f7f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle {n}}"></noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b6881107a598c20a60e72fe82bc41a4a1f7f4c" data-alt="{\displaystyle {n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  columns is called an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {m\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {m\times n}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76721b23ee82292ce7de5c1228513859661b51f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle {m\times n}}"></noscript><span class="lazy-image-placeholder" style="width: 6.276ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76721b23ee82292ce7de5c1228513859661b51f3" data-alt="{\displaystyle {m\times n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {m}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989d8ed6300d470470571f438bbe51f0693fb7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle {m}}"></noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989d8ed6300d470470571f438bbe51f0693fb7b2" data-alt="{\displaystyle {m}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> -by-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b6881107a598c20a60e72fe82bc41a4a1f7f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle {n}}"></noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b6881107a598c20a60e72fe82bc41a4a1f7f4c" data-alt="{\displaystyle {n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {m}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989d8ed6300d470470571f438bbe51f0693fb7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle {m}}"></noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989d8ed6300d470470571f438bbe51f0693fb7b2" data-alt="{\displaystyle {m}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b6881107a598c20a60e72fe82bc41a4a1f7f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle {n}}"></noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b6881107a598c20a60e72fe82bc41a4a1f7f4c" data-alt="{\displaystyle {n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are called its dimensions. For example, the matrix <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A} }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724d496b798f73a7173ef082565f5111ca547bb6" 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} }}"></noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724d496b798f73a7173ef082565f5111ca547bb6" data-alt="{\displaystyle {\mathbf {A} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  above is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {3\times 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>×</mo> <mn>2</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {3\times 2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f657f2e0de15cb9309ee4d1566b03ec938b03d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle {3\times 2}}"></noscript><span class="lazy-image-placeholder" style="width: 5.165ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f657f2e0de15cb9309ee4d1566b03ec938b03d45" data-alt="{\displaystyle {3\times 2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix. Matrices with a single row are called <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vectors</a>, and those with a single column are called <a href="/wiki/Column_vectors" class="mw-redirect" title="Column vectors">column vectors</a>. A matrix with the same number of rows and columns is called a <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a>.<a href="#cite_note-:4-5">[5]</a> A matrix with an infinite number of rows or columns (or both) is called an <a href="#Infinite_matrices"> infinite matrix</a>. In some contexts, such as <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra programs</a>, it is useful to consider a matrix with no rows or no columns, called an <a href="#Empty_matrix"> empty matrix</a>. <table class="wiki table"> <caption>Overview of a matrix size </caption> <tbody><tr> <th scope="col">Name </th> <th scope="col">Size </th> <th scope="col">Example </th> <th scope="col">Description </th> <th scope="col">Notation </th></tr> <tr> <th scope="row"><a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">Row vector</a> </th> <td>1 × n</td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}3&7&2\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}3&7&2\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47374c1d98d87b805e7c798d77865d060fc9780d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.178ex; height:2.843ex;" alt="{\displaystyle {\begin{bmatrix}3&7&2\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 10.178ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47374c1d98d87b805e7c798d77865d060fc9780d" data-alt="{\displaystyle {\begin{bmatrix}3&7&2\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td>A matrix with one row, sometimes used to represent a vector </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">{a_{i}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666961495bcc9330e949be5e4d0ddabacd6da633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{a_{i}}"></noscript><span class="lazy-image-placeholder" style="width: 2.029ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666961495bcc9330e949be5e4d0ddabacd6da633" data-alt="{a_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <th scope="col"><a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">Column vector</a> </th> <td>n × 1</td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}4\\1\\8\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}4\\1\\8\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f6732e6903bb84f36b682d8762f9ec5f8b489d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:5.015ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}4\\1\\8\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 5.015ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f6732e6903bb84f36b682d8762f9ec5f8b489d" data-alt="{\displaystyle {\begin{bmatrix}4\\1\\8\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td>A matrix with one column, sometimes used to represent a vector </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">{a_{j}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06eb52f2e6e616b5e99c5acf2d4c54023ee07dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{a_{j}}"></noscript><span class="lazy-image-placeholder" style="width: 2.14ex;height: 2.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06eb52f2e6e616b5e99c5acf2d4c54023ee07dc" data-alt="{a_{j}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <th scope="col"><a href="/wiki/Square_matrix" title="Square matrix">Square matrix</a> </th> <td>n × n</td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}9&13&5\\1&11&7\\2&6&3\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>9</mn> </mtd> <mtd> <mn>13</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>11</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}9&13&5\\1&11&7\\2&6&3\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55375914df4213b621f22cb1e5a0d6eb09af29df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.147ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}9&13&5\\1&11&7\\2&6&3\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 13.147ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55375914df4213b621f22cb1e5a0d6eb09af29df" data-alt="{\displaystyle {\begin{bmatrix}9&13&5\\1&11&7\\2&6&3\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td>A matrix with the same number of rows and columns, sometimes used to represent a <a href="#Linear_transformations">linear transformation</a> from a vector space to itself, such as <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a>, <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a>, or <a href="/wiki/Shear_mapping" title="Shear mapping">shearing</a>. </td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <annotation encoding="application/x-tex">{\mathbf {A} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a09c945b21828304a091e2d9706d13e876bd0d" 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="{\mathbf {A} }"></noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a09c945b21828304a091e2d9706d13e876bd0d" data-alt="{\mathbf {A} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr></tbody></table> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Notation">Notation</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=3" title="Edit section: Notation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written in <a href="/wiki/Square_bracket" class="mw-redirect" title="Square bracket">square brackets</a> or <a href="/wiki/Parentheses" class="mw-redirect" title="Parentheses">parentheses</a>, so that an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></noscript><span class="lazy-image-placeholder" style="width: 6.276ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" data-alt="{\displaystyle m\times n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix <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><noscript><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} }"></noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" data-alt="{\displaystyle \mathbf {A} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is represented as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5467918c1f72eed942e20d1c7662188194246f80" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:60.116ex; height:14.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 60.116ex;height: 14.176ex;vertical-align: -6.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5467918c1f72eed942e20d1c7662188194246f80" data-alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  This may be abbreviated by writing only a single generic term, possibly along with indices, as in <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} =\left(a_{ij}\right),\quad \left[a_{ij}\right],\quad {\text{or}}\quad \left(a_{ij}\right)_{1\leq i\leq m,\;1\leq j\leq n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <mrow> <mo>[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>]</mo> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em"></mspace> <msub> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>m</mi> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =\left(a_{ij}\right),\quad \left[a_{ij}\right],\quad {\text{or}}\quad \left(a_{ij}\right)_{1\leq i\leq m,\;1\leq j\leq n}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc2e9990806f2830d7a3865e6adb451a66e546c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:41.802ex; height:3.343ex;" alt="{\displaystyle \mathbf {A} =\left(a_{ij}\right),\quad \left[a_{ij}\right],\quad {\text{or}}\quad \left(a_{ij}\right)_{1\leq i\leq m,\;1\leq j\leq n}}"></noscript><span class="lazy-image-placeholder" style="width: 41.802ex;height: 3.343ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc2e9990806f2830d7a3865e6adb451a66e546c" data-alt="{\displaystyle \mathbf {A} =\left(a_{ij}\right),\quad \left[a_{ij}\right],\quad {\text{or}}\quad \left(a_{ij}\right)_{1\leq i\leq m,\;1\leq j\leq n}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} =(a_{i,j})_{1\leq i,j\leq n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =(a_{i,j})_{1\leq i,j\leq n}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e03f4cd8c02d54ca318a68dab549b3011837a0d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.391ex; height:3.009ex;" alt="{\displaystyle \mathbf {A} =(a_{i,j})_{1\leq i,j\leq n}}"></noscript><span class="lazy-image-placeholder" style="width: 16.391ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e03f4cd8c02d54ca318a68dab549b3011837a0d2" data-alt="{\displaystyle \mathbf {A} =(a_{i,j})_{1\leq i,j\leq n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in the case that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=m}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480d6131c6cb07a90f4ec18a376a59fab884b860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle n=m}"></noscript><span class="lazy-image-placeholder" style="width: 6.534ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480d6131c6cb07a90f4ec18a376a59fab884b860" data-alt="{\displaystyle n=m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Matrices are usually symbolized using <a href="/wiki/Upper-case" class="mw-redirect" title="Upper-case">upper-case</a> letters (such as <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A} }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724d496b798f73a7173ef082565f5111ca547bb6" 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} }}"></noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724d496b798f73a7173ef082565f5111ca547bb6" data-alt="{\displaystyle {\mathbf {A} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in the examples above), while the corresponding <a href="/wiki/Lower-case" class="mw-redirect" title="Lower-case">lower-case</a> letters, with two subscript indices (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{11}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{11}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a8aa9ca995d8bf721fc1be41447cf5ada51de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.106ex; height:2.009ex;" alt="{\displaystyle {a_{11}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.106ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a8aa9ca995d8bf721fc1be41447cf5ada51de8" data-alt="{\displaystyle {a_{11}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{1,1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{1,1}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4f8dec3f252168c3943d0ce7101c48b93a10546" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.563ex; height:2.343ex;" alt="{\displaystyle {a_{1,1}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.563ex;height: 2.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4f8dec3f252168c3943d0ce7101c48b93a10546" data-alt="{\displaystyle {a_{1,1}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special <a href="/wiki/Emphasis_(typography)" title="Emphasis (typography)">typographical style</a>, commonly boldface Roman (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a double-underline with the variable name, with or without boldface style, as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\underline {A}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <munder> <mi>A</mi> <mo>_</mo> </munder> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\underline {A}}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998c5de9803c6c1eaf3a4944fcdceed6f0af959f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.02ex; margin-bottom: -0.818ex; width:1.748ex; height:3.676ex;" alt="{\displaystyle {\underline {\underline {A}}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.748ex;height: 3.676ex;vertical-align: -1.02ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998c5de9803c6c1eaf3a4944fcdceed6f0af959f" data-alt="{\displaystyle {\underline {\underline {A}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The entry in the i-th row and j-th column of a matrix A is sometimes referred to as the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {i,j}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5876095735fdb75f51433429203661da7fca390f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle {i,j}}"></noscript><span class="lazy-image-placeholder" style="width: 2.794ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5876095735fdb75f51433429203661da7fca390f" data-alt="{\displaystyle {i,j}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {(i,j)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(i,j)}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a73d586334f9b11ebc9abee6a8fa7bcc01d7afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle {(i,j)}}"></noscript><span class="lazy-image-placeholder" style="width: 4.604ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a73d586334f9b11ebc9abee6a8fa7bcc01d7afd" data-alt="{\displaystyle {(i,j)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  entry of the matrix, and commonly denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{i,j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{i,j}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ce39a2b3dbcebc3eddfd54e07da2154858868e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.343ex;" alt="{\displaystyle {a_{i,j}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.164ex;height: 2.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ce39a2b3dbcebc3eddfd54e07da2154858868e" data-alt="{\displaystyle {a_{i,j}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{ij}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{ij}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eae96c51109ab849da6fafb657ec794631abd37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle {a_{ij}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.707ex;height: 2.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eae96c51109ab849da6fafb657ec794631abd37" data-alt="{\displaystyle {a_{ij}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Alternative notations for that entry are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {A} [i,j]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A} [i,j]}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb3630aa110e0aa90d0bc0e4c0b55f659d3d63e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.108ex; height:2.843ex;" alt="{\displaystyle {\mathbf {A} [i,j]}}"></noscript><span class="lazy-image-placeholder" style="width: 6.108ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb3630aa110e0aa90d0bc0e4c0b55f659d3d63e" data-alt="{\displaystyle {\mathbf {A} [i,j]}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {A} _{i,j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A} _{i,j}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a8257645480f71cc7d9c081fe0a45dff9a31ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.954ex; height:2.843ex;" alt="{\displaystyle {\mathbf {A} _{i,j}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.954ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a8257645480f71cc7d9c081fe0a45dff9a31ec1" data-alt="{\displaystyle {\mathbf {A} _{i,j}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . For example, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,3)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c80e593e3953231c56d0887f5b247bbe517461f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,3)}"></noscript><span class="lazy-image-placeholder" style="width: 5.168ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c80e593e3953231c56d0887f5b247bbe517461f" data-alt="{\displaystyle (1,3)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  entry of the following matrix <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><noscript><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} }"></noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" data-alt="{\displaystyle \mathbf {A} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is 5 (also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{13}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{13}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff4257b367283e6ecdf4af184f8355c0c2b4d34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.106ex; height:2.009ex;" alt="{\displaystyle {a_{13}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.106ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff4257b367283e6ecdf4af184f8355c0c2b4d34" data-alt="{\displaystyle {a_{13}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {a_{1,3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{1,3}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb06a058031e3f07851a7ee7ee31140021cb0a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.563ex; height:2.343ex;" alt="{\displaystyle {a_{1,3}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.563ex;height: 2.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb06a058031e3f07851a7ee7ee31140021cb0a06" data-alt="{\displaystyle {a_{1,3}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} [1,3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} [1,3]}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3b9da80148fbbfe012952941a2bdc0d8393c38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.672ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} [1,3]}"></noscript><span class="lazy-image-placeholder" style="width: 6.672ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3b9da80148fbbfe012952941a2bdc0d8393c38" data-alt="{\displaystyle \mathbf {A} [1,3]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{\mathbf {A} }_{1,3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{\mathbf {A} }_{1,3}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e49001b42606ad3da547f8d6efa13e48d9ed1dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.353ex; height:2.843ex;" alt="{\displaystyle {{\mathbf {A} }_{1,3}}}"></noscript><span class="lazy-image-placeholder" style="width: 4.353ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e49001b42606ad3da547f8d6efa13e48d9ed1dd" data-alt="{\displaystyle {{\mathbf {A} }_{1,3}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{bmatrix}4&-7&\color {red}{5}&0\\-2&0&11&8\\19&1&-3&12\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mo>−</mo> <mn>7</mn> </mtd> <mtd> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mstyle> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>11</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>19</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>12</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}4&-7&\color {red}{5}&0\\-2&0&11&8\\19&1&-3&12\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d58776afc906466b6163beaf584405eaf7a97f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:27.175ex; height:9.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}4&-7&\color {red}{5}&0\\-2&0&11&8\\19&1&-3&12\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 27.175ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d58776afc906466b6163beaf584405eaf7a97f" data-alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}4&-7&\color {red}{5}&0\\-2&0&11&8\\19&1&-3&12\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Sometimes, the entries of a matrix can be defined by a formula such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,j}=f(i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,j}=f(i,j)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4451079309af20865d76b996b62f06fbe41314e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.145ex; height:3.009ex;" alt="{\displaystyle a_{i,j}=f(i,j)}"></noscript><span class="lazy-image-placeholder" style="width: 12.145ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4451079309af20865d76b996b62f06fbe41314e" data-alt="{\displaystyle a_{i,j}=f(i,j)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . For example, each of the entries of the following matrix <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><noscript><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} }"></noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" data-alt="{\displaystyle \mathbf {A} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is determined by the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}=i-j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mo>−</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}=i-j}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d745dd14d76ec708e0c984bf2489509caecd2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.406ex; height:2.843ex;" alt="{\displaystyle a_{ij}=i-j}"></noscript><span class="lazy-image-placeholder" style="width: 10.406ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d745dd14d76ec708e0c984bf2489509caecd2c" data-alt="{\displaystyle a_{ij}=i-j}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{bmatrix}0&-1&-2&-3\\1&0&-1&-2\\2&1&0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}0&-1&-2&-3\\1&0&-1&-2\\2&1&0&-1\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f51de53bfecb43be29e3905aad271db49679f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:26.012ex; height:9.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}0&-1&-2&-3\\1&0&-1&-2\\2&1&0&-1\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 26.012ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f51de53bfecb43be29e3905aad271db49679f50" data-alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}0&-1&-2&-3\\1&0&-1&-2\\2&1&0&-1\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parentheses. For example, the matrix above is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {A} }=[i-j]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>i</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A} }=[i-j]}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9724402762df013fda41e49ca5572dc9b89cd9cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.013ex; height:2.843ex;" alt="{\displaystyle {\mathbf {A} }=[i-j]}"></noscript><span class="lazy-image-placeholder" style="width: 11.013ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9724402762df013fda41e49ca5572dc9b89cd9cc" data-alt="{\displaystyle {\mathbf {A} }=[i-j]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {A} }=((i-j))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A} }=((i-j))}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502ff9ccab5be63b40f2c8901ab77cdf15d45712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.337ex; height:2.843ex;" alt="{\displaystyle {\mathbf {A} }=((i-j))}"></noscript><span class="lazy-image-placeholder" style="width: 13.337ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502ff9ccab5be63b40f2c8901ab77cdf15d45712" data-alt="{\displaystyle {\mathbf {A} }=((i-j))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . If matrix size is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></noscript><span class="lazy-image-placeholder" style="width: 6.276ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" data-alt="{\displaystyle m\times n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the above-mentioned formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(i,j)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb5444a99e0ebb8684c4544152ab1268160da20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.882ex; height:2.843ex;" alt="{\displaystyle f(i,j)}"></noscript><span class="lazy-image-placeholder" style="width: 5.882ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb5444a99e0ebb8684c4544152ab1268160da20" data-alt="{\displaystyle f(i,j)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is valid for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dots ,m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dots ,m}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d38508db1c01a8fbea6da93f1866f86944e3f12d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.282ex; height:2.509ex;" alt="{\displaystyle i=1,\dots ,m}"></noscript><span class="lazy-image-placeholder" style="width: 12.282ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d38508db1c01a8fbea6da93f1866f86944e3f12d" data-alt="{\displaystyle i=1,\dots ,m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=1,\dots ,n}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f9393eaa189f1fb2c747b687b7b8d67640d5f1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:11.819ex; height:2.509ex;" alt="{\displaystyle j=1,\dots ,n}"></noscript><span class="lazy-image-placeholder" style="width: 11.819ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f9393eaa189f1fb2c747b687b7b8d67640d5f1e" data-alt="{\displaystyle j=1,\dots ,n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This can be specified separately or indicated using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></noscript><span class="lazy-image-placeholder" style="width: 6.276ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" data-alt="{\displaystyle m\times n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as a subscript. For instance, the matrix <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><noscript><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} }"></noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" data-alt="{\displaystyle \mathbf {A} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  above is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 4}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fda443e7a6e78fa880a6dccbf8bdf43a10d9988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 4}"></noscript><span class="lazy-image-placeholder" style="width: 5.165ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fda443e7a6e78fa880a6dccbf8bdf43a10d9988" data-alt="{\displaystyle 3\times 4}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and can be defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {A} }=[i-j](i=1,2,3;j=1,\dots ,4)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>i</mi> <mo>−</mo> <mi>j</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>;</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A} }=[i-j](i=1,2,3;j=1,\dots ,4)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad68413c2769db2a39b51a017fb2e45cd793d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.872ex; height:2.843ex;" alt="{\displaystyle {\mathbf {A} }=[i-j](i=1,2,3;j=1,\dots ,4)}"></noscript><span class="lazy-image-placeholder" style="width: 34.872ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad68413c2769db2a39b51a017fb2e45cd793d3b" data-alt="{\displaystyle {\mathbf {A} }=[i-j](i=1,2,3;j=1,\dots ,4)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {A} }=[i-j]_{3\times 4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>i</mi> <mo>−</mo> <mi>j</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>×</mo> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A} }=[i-j]_{3\times 4}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ab376d25a953520208baf764ea114a1ee565b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.167ex; height:2.843ex;" alt="{\displaystyle {\mathbf {A} }=[i-j]_{3\times 4}}"></noscript><span class="lazy-image-placeholder" style="width: 14.167ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ab376d25a953520208baf764ea114a1ee565b9" data-alt="{\displaystyle {\mathbf {A} }=[i-j]_{3\times 4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an m-by-n matrix. Some programming languages start the numbering of array indexes at zero, in which case the entries of an m-by-n matrix are indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq i\leq m-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq i\leq m-1}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4b2f05ce28398ed53128680921025e92b3fc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.205ex; height:2.343ex;" alt="{\displaystyle 0\leq i\leq m-1}"></noscript><span class="lazy-image-placeholder" style="width: 14.205ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4b2f05ce28398ed53128680921025e92b3fc4e" data-alt="{\displaystyle 0\leq i\leq m-1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq j\leq n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq j\leq n-1}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a80ba06f8c7cb92e65b1932f837ceb3e9a405e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.715ex; height:2.509ex;" alt="{\displaystyle 0\leq j\leq n-1}"></noscript><span class="lazy-image-placeholder" style="width: 13.715ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a80ba06f8c7cb92e65b1932f837ceb3e9a405e51" data-alt="{\displaystyle 0\leq j\leq n-1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .<a href="#cite_note-6">[6]</a> This article follows the more common convention in mathematical writing where enumeration starts from 1. The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all m-by-n real matrices is often denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}(m,n),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}(m,n),}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ef01d32be11aa8c6420356b7d17a6233b134ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.716ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}(m,n),}"></noscript><span class="lazy-image-placeholder" style="width: 9.716ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ef01d32be11aa8c6420356b7d17a6233b134ce" data-alt="{\displaystyle {\mathcal {M}}(m,n),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{m\times n}(\mathbb {R} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{m\times n}(\mathbb {R} ).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b72ca9be1664df5941623f3faacf9a45225bb07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.865ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{m\times n}(\mathbb {R} ).}"></noscript><span class="lazy-image-placeholder" style="width: 10.865ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b72ca9be1664df5941623f3faacf9a45225bb07" data-alt="{\displaystyle {\mathcal {M}}_{m\times n}(\mathbb {R} ).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  The set of all m-by-n matrices over another <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, or over a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> R, is similarly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}(m,n,R),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}(m,n,R),}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e466405aaaf3c8009e3404fe51b31c6bd70448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.514ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}(m,n,R),}"></noscript><span class="lazy-image-placeholder" style="width: 12.514ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e466405aaaf3c8009e3404fe51b31c6bd70448" data-alt="{\displaystyle {\mathcal {M}}(m,n,R),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{m\times n}(R).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{m\times n}(R).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6640d032642fde9cae2e4025ad394382b66a27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.951ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{m\times n}(R).}"></noscript><span class="lazy-image-placeholder" style="width: 10.951ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6640d032642fde9cae2e4025ad394382b66a27" data-alt="{\displaystyle {\mathcal {M}}_{m\times n}(R).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  If m = n, such as in the case of <a href="/wiki/Square_matrices" class="mw-redirect" title="Square matrices">square matrices</a>, one does not repeat the dimension: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}(n,R),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}(n,R),}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f410ab3e0df4825b08083b93949bb36d82d32536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.439ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}(n,R),}"></noscript><span class="lazy-image-placeholder" style="width: 9.439ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f410ab3e0df4825b08083b93949bb36d82d32536" data-alt="{\displaystyle {\mathcal {M}}(n,R),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{n}(R).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{n}(R).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53681a5d65e9480980a5cc7e66ddfa16ecea430c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.229ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}_{n}(R).}"></noscript><span class="lazy-image-placeholder" style="width: 8.229ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53681a5d65e9480980a5cc7e66ddfa16ecea430c" data-alt="{\displaystyle {\mathcal {M}}_{n}(R).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> <a href="#cite_note-Pop-Furdui-7">[7]</a> Often, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Mat} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42fdf1777541a3ef4e694461fa576f1aabe4375f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.198ex; height:2.176ex;" alt="{\displaystyle \operatorname {Mat} }"></noscript><span class="lazy-image-placeholder" style="width: 4.198ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42fdf1777541a3ef4e694461fa576f1aabe4375f" data-alt="{\displaystyle \operatorname {Mat} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , is used in place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101831116f6abff5a4a6d5f21af9b643bd0bd865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.437ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}.}"></noscript><span class="lazy-image-placeholder" style="width: 3.437ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101831116f6abff5a4a6d5f21af9b643bd0bd865" data-alt="{\displaystyle {\mathcal {M}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Basic_operations">Basic operations</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=4" title="Edit section: Basic operations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> Several basic operations can be applied to matrices. Some, such as transposition and submatrix do not depend on the nature of the entries. Others, such as matrix addition, scalar multiplication, matrix multiplication, and row operations involve operations on matrix entries and therefore require that matrix entries are numbers or belong to a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> or a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>.<a href="#cite_note-8">[8]</a> In this section, it is supposed that matrix entries belong to a fixed ring, which is typically a field of numbers. <div class="mw-heading mw-heading3"><h3 id="Addition,_scalar_multiplication,_subtraction_and_transposition">Addition, scalar multiplication, subtraction and transposition</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=5" title="Edit section: Addition, scalar multiplication, subtraction and transposition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <dl><dt><a href="/wiki/Matrix_addition" title="Matrix addition">Addition</a></dt></dl> The sum A + B of two m×n matrices A and B is calculated entrywise: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathbf {A}}+{\mathbf {B}})_{i,j}={\mathbf {A}}_{i,j}+{\mathbf {B}}_{i,j},\quad 1\leq i\leq m,\quad 1\leq j\leq n.}"> <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"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em"></mspace> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>m</mi> <mo>,</mo> <mspace width="1em"></mspace> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathbf {A}}+{\mathbf {B}})_{i,j}={\mathbf {A}}_{i,j}+{\mathbf {B}}_{i,j},\quad 1\leq i\leq m,\quad 1\leq j\leq n.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa806786f57e2ead8e350d89213a682bc2f1d2a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.507ex; height:3.009ex;" alt="{\displaystyle ({\mathbf {A}}+{\mathbf {B}})_{i,j}={\mathbf {A}}_{i,j}+{\mathbf {B}}_{i,j},\quad 1\leq i\leq m,\quad 1\leq j\leq n.}"></noscript><span class="lazy-image-placeholder" style="width: 51.507ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa806786f57e2ead8e350d89213a682bc2f1d2a3" data-alt="{\displaystyle ({\mathbf {A}}+{\mathbf {B}})_{i,j}={\mathbf {A}}_{i,j}+{\mathbf {B}}_{i,j},\quad 1\leq i\leq m,\quad 1\leq j\leq n.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For example, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&3&1\\1&0&0\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\end{bmatrix}}={\begin{bmatrix}1+0&3+0&1+5\\1+7&0+5&0+0\end{bmatrix}}={\begin{bmatrix}1&3&6\\8&5&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <mn>0</mn> </mtd> <mtd> <mn>3</mn> <mo>+</mo> <mn>0</mn> </mtd> <mtd> <mn>1</mn> <mo>+</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <mn>7</mn> </mtd> <mtd> <mn>0</mn> <mo>+</mo> <mn>5</mn> </mtd> <mtd> <mn>0</mn> <mo>+</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&3&1\\1&0&0\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\end{bmatrix}}={\begin{bmatrix}1+0&3+0&1+5\\1+7&0+5&0+0\end{bmatrix}}={\begin{bmatrix}1&3&6\\8&5&0\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e600aa691a93ceb33f0fdac290002d1391d6688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.402ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&3&1\\1&0&0\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\end{bmatrix}}={\begin{bmatrix}1+0&3+0&1+5\\1+7&0+5&0+0\end{bmatrix}}={\begin{bmatrix}1&3&6\\8&5&0\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 66.402ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e600aa691a93ceb33f0fdac290002d1391d6688" data-alt="{\displaystyle {\begin{bmatrix}1&3&1\\1&0&0\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\end{bmatrix}}={\begin{bmatrix}1+0&3+0&1+5\\1+7&0+5&0+0\end{bmatrix}}={\begin{bmatrix}1&3&6\\8&5&0\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd> <dt><a href="/wiki/Scalar_multiplication" title="Scalar multiplication">Scalar multiplication</a></dt></dl> The product cA of a number c (also called a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> in this context) and a matrix A is computed by multiplying every entry of A by c: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c{\mathbf {A}})_{i,j}=c\cdot {\mathbf {A}}_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>c</mi> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c{\mathbf {A}})_{i,j}=c\cdot {\mathbf {A}}_{i,j}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99dc53991b812afcf155f2e546cd41a6ae36f10" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.509ex; height:3.009ex;" alt="{\displaystyle (c{\mathbf {A}})_{i,j}=c\cdot {\mathbf {A}}_{i,j}}"></noscript><span class="lazy-image-placeholder" style="width: 16.509ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99dc53991b812afcf155f2e546cd41a6ae36f10" data-alt="{\displaystyle (c{\mathbf {A}})_{i,j}=c\cdot {\mathbf {A}}_{i,j}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  This operation is called scalar multiplication, but its result is not named "scalar product" to avoid confusion, since "scalar product" is often used as a synonym for "<a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a>". For example: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\cdot {\begin{bmatrix}1&8&-3\\4&-2&5\end{bmatrix}}={\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot -3\\2\cdot 4&2\cdot -2&2\cdot 5\end{bmatrix}}={\begin{bmatrix}2&16&-6\\8&-4&10\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>8</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mn>10</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot {\begin{bmatrix}1&8&-3\\4&-2&5\end{bmatrix}}={\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot -3\\2\cdot 4&2\cdot -2&2\cdot 5\end{bmatrix}}={\begin{bmatrix}2&16&-6\\8&-4&10\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7fe1075c8c575225e74096007bef9205c88964" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.429ex; height:6.176ex;" alt="{\displaystyle 2\cdot {\begin{bmatrix}1&8&-3\\4&-2&5\end{bmatrix}}={\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot -3\\2\cdot 4&2\cdot -2&2\cdot 5\end{bmatrix}}={\begin{bmatrix}2&16&-6\\8&-4&10\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 62.429ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7fe1075c8c575225e74096007bef9205c88964" data-alt="{\displaystyle 2\cdot {\begin{bmatrix}1&8&-3\\4&-2&5\end{bmatrix}}={\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot -3\\2\cdot 4&2\cdot -2&2\cdot 5\end{bmatrix}}={\begin{bmatrix}2&16&-6\\8&-4&10\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd> <dt><a href="/wiki/Matrix_subtraction" class="mw-redirect" title="Matrix subtraction">Subtraction</a></dt></dl> The subtraction of two m×n matrices is defined by composing matrix addition with scalar multiplication by –1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} -\mathbf {B} =\mathbf {A} +(-1)\cdot \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} -\mathbf {B} =\mathbf {A} +(-1)\cdot \mathbf {B} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1743905b472434f9b2ffec5c159062c1b77e9643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.079ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} -\mathbf {B} =\mathbf {A} +(-1)\cdot \mathbf {B} }"></noscript><span class="lazy-image-placeholder" style="width: 23.079ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1743905b472434f9b2ffec5c159062c1b77e9643" data-alt="{\displaystyle \mathbf {A} -\mathbf {B} =\mathbf {A} +(-1)\cdot \mathbf {B} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd> <dt><a href="/wiki/Transpose" title="Transpose">Transposition</a></dt></dl> The transpose of an m×n matrix A is the n×m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\mathbf {A}}^{\rm {T}}\right)_{i,j}={\mathbf {A}}_{j,i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\mathbf {A}}^{\rm {T}}\right)_{i,j}={\mathbf {A}}_{j,i}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a593edc189c58430468491beba276358a71a8761" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:15.202ex; height:3.843ex;" alt="{\displaystyle \left({\mathbf {A}}^{\rm {T}}\right)_{i,j}={\mathbf {A}}_{j,i}.}"></noscript><span class="lazy-image-placeholder" style="width: 15.202ex;height: 3.843ex;vertical-align: -1.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a593edc189c58430468491beba276358a71a8761" data-alt="{\displaystyle \left({\mathbf {A}}^{\rm {T}}\right)_{i,j}={\mathbf {A}}_{j,i}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For example: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&3\\0&-6&7\end{bmatrix}}^{\mathrm {T} }={\begin{bmatrix}1&0\\2&-6\\3&7\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&3\\0&-6&7\end{bmatrix}}^{\mathrm {T} }={\begin{bmatrix}1&0\\2&-6\\3&7\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51f6dba024e104b412ed0562163ca9a11fcb9463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:27.972ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&3\\0&-6&7\end{bmatrix}}^{\mathrm {T} }={\begin{bmatrix}1&0\\2&-6\\3&7\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 27.972ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51f6dba024e104b412ed0562163ca9a11fcb9463" data-alt="{\displaystyle {\begin{bmatrix}1&2&3\\0&-6&7\end{bmatrix}}^{\mathrm {T} }={\begin{bmatrix}1&0\\2&-6\\3&7\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Familiar properties of numbers extend to these operations on matrices: for example, addition is <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, that is, the matrix sum does not depend on the order of the summands: A + B = B + A.<a href="#cite_note-9">[9]</a> The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT. Finally, (AT)T = A. <div class="mw-heading mw-heading3"><h3 id="Matrix_multiplication">Matrix multiplication</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=6" title="Edit section: Matrix multiplication" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 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/Matrix_multiplication" title="Matrix multiplication">Matrix multiplication</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:MatrixMultiplication.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/MatrixMultiplication.png/300px-MatrixMultiplication.png" decoding="async" width="300" height="180" class="mw-file-element" data-file-width="1000" data-file-height="599"></noscript><span class="lazy-image-placeholder" style="width: 300px;height: 180px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/MatrixMultiplication.png/300px-MatrixMultiplication.png" data-width="300" data-height="180" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/MatrixMultiplication.png/450px-MatrixMultiplication.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/MatrixMultiplication.png/600px-MatrixMultiplication.png 2x" data-class="mw-file-element"> </a><figcaption>Schematic depiction of the matrix product AB of two matrices A and B</figcaption></figure> Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m×n matrix and B is an n×p matrix, then their matrix product AB is the m×p matrix whose entries are given by <a href="/wiki/Dot_product" title="Dot product">dot product</a> of the corresponding row of A and the corresponding column of B:<a href="#cite_note-:5-10">[10]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbf {AB} ]_{i,j}=a_{i,1}b_{1,j}+a_{i,2}b_{2,j}+\cdots +a_{i,n}b_{n,j}=\sum _{r=1}^{n}a_{i,r}b_{r,j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>r</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mathbf {AB} ]_{i,j}=a_{i,1}b_{1,j}+a_{i,2}b_{2,j}+\cdots +a_{i,n}b_{n,j}=\sum _{r=1}^{n}a_{i,r}b_{r,j},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c903c2c14d249005ce9ebaa47a8d6c6710c1c29e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.128ex; height:6.843ex;" alt="{\displaystyle [\mathbf {AB} ]_{i,j}=a_{i,1}b_{1,j}+a_{i,2}b_{2,j}+\cdots +a_{i,n}b_{n,j}=\sum _{r=1}^{n}a_{i,r}b_{r,j},}"></noscript><span class="lazy-image-placeholder" style="width: 55.128ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c903c2c14d249005ce9ebaa47a8d6c6710c1c29e" data-alt="{\displaystyle [\mathbf {AB} ]_{i,j}=a_{i,1}b_{1,j}+a_{i,2}b_{2,j}+\cdots +a_{i,n}b_{n,j}=\sum _{r=1}^{n}a_{i,r}b_{r,j},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where 1 ≤ i ≤ m and 1 ≤ j ≤ p.<a href="#cite_note-11">[11]</a> For example, the underlined entry 2340 in the product is calculated as (2 × 1000) + (3 × 100) + (4 × 10) = 2340: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\begin{bmatrix}{\underline {2}}&{\underline {3}}&{\underline {4}}\\1&0&0\\\end{bmatrix}}{\begin{bmatrix}0&{\underline {1000}}\\1&{\underline {100}}\\0&{\underline {10}}\\\end{bmatrix}}&={\begin{bmatrix}3&{\underline {2340}}\\0&1000\\\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"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>2</mn> <mo>_</mo> </munder> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>3</mn> <mo>_</mo> </munder> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>4</mn> <mo>_</mo> </munder> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>1000</mn> <mo>_</mo> </munder> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>100</mn> <mo>_</mo> </munder> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>10</mn> <mo>_</mo> </munder> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <munder> <mn>2340</mn> <mo>_</mo> </munder> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1000</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{bmatrix}{\underline {2}}&{\underline {3}}&{\underline {4}}\\1&0&0\\\end{bmatrix}}{\begin{bmatrix}0&{\underline {1000}}\\1&{\underline {100}}\\0&{\underline {10}}\\\end{bmatrix}}&={\begin{bmatrix}3&{\underline {2340}}\\0&1000\\\end{bmatrix}}.\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8435ba88efca5a73e7d1b122bb19f99ef136d71e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.979ex; margin-bottom: -0.859ex; width:39.169ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}{\begin{bmatrix}{\underline {2}}&{\underline {3}}&{\underline {4}}\\1&0&0\\\end{bmatrix}}{\begin{bmatrix}0&{\underline {1000}}\\1&{\underline {100}}\\0&{\underline {10}}\\\end{bmatrix}}&={\begin{bmatrix}3&{\underline {2340}}\\0&1000\\\end{bmatrix}}.\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 39.169ex;height: 10.176ex;vertical-align: -3.979ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8435ba88efca5a73e7d1b122bb19f99ef136d71e" data-alt="{\displaystyle {\begin{aligned}{\begin{bmatrix}{\underline {2}}&{\underline {3}}&{\underline {4}}\\1&0&0\\\end{bmatrix}}{\begin{bmatrix}0&{\underline {1000}}\\1&{\underline {100}}\\0&{\underline {10}}\\\end{bmatrix}}&={\begin{bmatrix}3&{\underline {2340}}\\0&1000\\\end{bmatrix}}.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Matrix multiplication satisfies the rules (AB)C = A(BC) (<a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a>), and (A + B)C = AC + BC as well as C(A + B) = CA + CB (left and right <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a>), whenever the size of the matrices is such that the various products are defined.<a href="#cite_note-12">[12]</a> The product AB may be defined without BA being defined, namely if A and B are m×n and n×k matrices, respectively, and m ≠ k. Even if both products are defined, they generally need not be equal, that is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {AB}}\neq {\mathbf {BA}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {AB}}\neq {\mathbf {BA}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c668f86bb3696952f5ff52998df9d0a6ba8c73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.586ex; height:2.676ex;" alt="{\displaystyle {\mathbf {AB}}\neq {\mathbf {BA}}.}"></noscript><span class="lazy-image-placeholder" style="width: 11.586ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c668f86bb3696952f5ff52998df9d0a6ba8c73" data-alt="{\displaystyle {\mathbf {AB}}\neq {\mathbf {BA}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  In other words, matrix multiplication is not <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>, in marked contrast to (rational, real, or complex) numbers, whose product is independent of the order of the factors.<a href="#cite_note-:5-10">[10]</a> An example of two matrices not commuting with each other is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}={\begin{bmatrix}0&1\\0&3\\\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}={\begin{bmatrix}0&1\\0&3\\\end{bmatrix}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f04b29d8a273a3d2ccff8b2ac98c9e75c305c7bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.307ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}={\begin{bmatrix}0&1\\0&3\\\end{bmatrix}},}"></noscript><span class="lazy-image-placeholder" style="width: 27.307ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f04b29d8a273a3d2ccff8b2ac98c9e75c305c7bd" data-alt="{\displaystyle {\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}={\begin{bmatrix}0&1\\0&3\\\end{bmatrix}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> whereas <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}={\begin{bmatrix}3&4\\0&0\\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}={\begin{bmatrix}3&4\\0&0\\\end{bmatrix}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d260e7e7dda974d50647f542f9968c9267eff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.307ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}={\begin{bmatrix}3&4\\0&0\\\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 27.307ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d260e7e7dda974d50647f542f9968c9267eff0" data-alt="{\displaystyle {\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}={\begin{bmatrix}3&4\\0&0\\\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Besides the ordinary matrix multiplication just described, other less frequently used operations on matrices that can be considered forms of multiplication also exist, such as the <a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a> and the <a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a>.<a href="#cite_note-13">[13]</a> They arise in solving matrix equations such as the <a href="/wiki/Sylvester_equation" title="Sylvester equation">Sylvester equation</a>. <div class="mw-heading mw-heading3"><h3 id="Row_operations">Row operations</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=7" title="Edit section: Row operations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 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/Row_operations" class="mw-redirect" title="Row operations">Row operations</a></div> There are three types of row operations: <ol><li>row addition, that is adding a row to another.</li> <li>row multiplication, that is multiplying all entries of a row by a non-zero constant;</li> <li>row switching, that is interchanging two rows of a matrix;</li></ol> These operations are used in several ways, including solving <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a> and finding <a href="/wiki/Matrix_inverse" class="mw-redirect" title="Matrix inverse">matrix inverses</a>. <div class="mw-heading mw-heading3"><h3 id="Submatrix"> Submatrix</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=8" title="Edit section: Submatrix" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> A submatrix of a matrix is a matrix obtained by deleting any collection of rows and/or columns.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a><a href="#cite_note-Protter_1970_869-16">[16]</a> For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{bmatrix}1&\color {red}{2}&3&4\\5&\color {red}{6}&7&8\\\color {red}{9}&\color {red}{10}&\color {red}{11}&\color {red}{12}\end{bmatrix}}\rightarrow {\begin{bmatrix}1&3&4\\5&7&8\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mstyle> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </mstyle> </mtd> <mtd> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </mstyle> </mtd> <mtd> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </mstyle> </mtd> <mtd> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mstyle> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}1&\color {red}{2}&3&4\\5&\color {red}{6}&7&8\\\color {red}{9}&\color {red}{10}&\color {red}{11}&\color {red}{12}\end{bmatrix}}\rightarrow {\begin{bmatrix}1&3&4\\5&7&8\end{bmatrix}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7347ef1fbe8802dde70d98638277d3ca81ec4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:39.675ex; height:9.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}1&\color {red}{2}&3&4\\5&\color {red}{6}&7&8\\\color {red}{9}&\color {red}{10}&\color {red}{11}&\color {red}{12}\end{bmatrix}}\rightarrow {\begin{bmatrix}1&3&4\\5&7&8\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 39.675ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7347ef1fbe8802dde70d98638277d3ca81ec4e" data-alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}1&\color {red}{2}&3&4\\5&\color {red}{6}&7&8\\\color {red}{9}&\color {red}{10}&\color {red}{11}&\color {red}{12}\end{bmatrix}}\rightarrow {\begin{bmatrix}1&3&4\\5&7&8\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> The <a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">minors</a> and cofactors of a matrix are found by computing the <a href="/wiki/Determinant" title="Determinant">determinant</a> of certain submatrices.<a href="#cite_note-Protter_1970_869-16">[16]</a><a href="#cite_note-17">[17]</a> A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain.<a href="#cite_note-18">[18]</a><a href="#cite_note-19">[19]</a> Other authors define a principal submatrix as one in which the first k rows and columns, for some number k, are the ones that remain;<a href="#cite_note-20">[20]</a> this type of submatrix has also been called a leading principal submatrix.<a href="#cite_note-21">[21]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Linear_equations">Linear equations</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=9" title="Edit section: Linear equations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Linear_equation" title="Linear equation">Linear equation</a> and <a href="/wiki/System_of_linear_equations" title="System of linear equations">System of linear equations</a></div> Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an m×n matrix, x designates a column vector (that is, n×1-matrix) of n variables x1, x2, ..., xn, and b is an m×1-column vector, then the matrix equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Ax} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Ax} =\mathbf {b} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e68388b7df59f536a3bef4e70def2f2bb36f48c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {Ax} =\mathbf {b} }"></noscript><span class="lazy-image-placeholder" style="width: 8.014ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e68388b7df59f536a3bef4e70def2f2bb36f48c0" data-alt="{\displaystyle \mathbf {Ax} =\mathbf {b} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> is equivalent to the system of linear equations<a href="#cite_note-22">[22]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a_{1,1}x_{1}+a_{1,2}x_{2}+&\cdots +a_{1,n}x_{n}=b_{1}\\&\ \ \vdots \\a_{m,1}x_{1}+a_{m,2}x_{2}+&\cdots +a_{m,n}x_{n}=b_{m}\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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> </mtd> <mtd> <mi></mi> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mtext> </mtext> <mtext> </mtext> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> </mtd> <mtd> <mi></mi> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a_{1,1}x_{1}+a_{1,2}x_{2}+&\cdots +a_{1,n}x_{n}=b_{1}\\&\ \ \vdots \\a_{m,1}x_{1}+a_{m,2}x_{2}+&\cdots +a_{m,n}x_{n}=b_{m}\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d36435b1869be0a1174a28d251c44929556218" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.388ex; margin-bottom: -0.284ex; width:37.155ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}a_{1,1}x_{1}+a_{1,2}x_{2}+&\cdots +a_{1,n}x_{n}=b_{1}\\&\ \ \vdots \\a_{m,1}x_{1}+a_{m,2}x_{2}+&\cdots +a_{m,n}x_{n}=b_{m}\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 37.155ex;height: 10.509ex;vertical-align: -4.388ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d36435b1869be0a1174a28d251c44929556218" data-alt="{\displaystyle {\begin{aligned}a_{1,1}x_{1}+a_{1,2}x_{2}+&\cdots +a_{1,n}x_{n}=b_{1}\\&\ \ \vdots \\a_{m,1}x_{1}+a_{m,2}x_{2}+&\cdots +a_{m,n}x_{n}=b_{m}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If n = m and the equations are <a href="/wiki/Independent_equation" title="Independent equation">independent</a>, then this can be done by writing <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\mathbf {A} ^{-1}\mathbf {b} }"> <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"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\mathbf {A} ^{-1}\mathbf {b} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079fdc93cc44d2d3d6bdf92dadb0ad8ade2efe09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.347ex; height:2.676ex;" alt="{\displaystyle \mathbf {x} =\mathbf {A} ^{-1}\mathbf {b} }"></noscript><span class="lazy-image-placeholder" style="width: 10.347ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/079fdc93cc44d2d3d6bdf92dadb0ad8ade2efe09" data-alt="{\displaystyle \mathbf {x} =\mathbf {A} ^{-1}\mathbf {b} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where A−1 is the <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse matrix</a> of A. If A has no inverse, solutions—if any—can be found using its <a href="/wiki/Generalized_inverse" title="Generalized inverse">generalized inverse</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Linear_transformations">Linear transformations</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=10" title="Edit section: Linear transformations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">Linear transformation</a> and <a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation matrix</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Area_parallellogram_as_determinant.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Area_parallellogram_as_determinant.svg/220px-Area_parallellogram_as_determinant.svg.png" decoding="async" width="220" height="253" class="mw-file-element" data-file-width="870" data-file-height="1000"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 253px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Area_parallellogram_as_determinant.svg/220px-Area_parallellogram_as_determinant.svg.png" data-width="220" data-height="253" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Area_parallellogram_as_determinant.svg/330px-Area_parallellogram_as_determinant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Area_parallellogram_as_determinant.svg/440px-Area_parallellogram_as_determinant.svg.png 2x" data-class="mw-file-element"> </a><figcaption>The vectors represented by a 2-by-2 matrix correspond to the sides of a unit square transformed into a parallelogram.</figcaption></figure> Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real m-by-n matrix A gives rise to a linear transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cccdefd5f0e00fc2fa5fde2d8cbb039cc408a035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.864ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}"></noscript><span class="lazy-image-placeholder" style="width: 9.864ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cccdefd5f0e00fc2fa5fde2d8cbb039cc408a035" data-alt="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  mapping each vector x in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></noscript><span class="lazy-image-placeholder" style="width: 2.897ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" data-alt="{\displaystyle \mathbb {R} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠ to the (matrix) product Ax, which is a vector in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{m}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/300954c24584ecb8256b59b1d1e6ea54ae1ae889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}.}"></noscript><span class="lazy-image-placeholder" style="width: 4ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/300954c24584ecb8256b59b1d1e6ea54ae1ae889" data-alt="{\displaystyle \mathbb {R} ^{m}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠ Conversely, each linear transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad78382c3d23bcb4051b3148f1a23b1d0ba52e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.079ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}"></noscript><span class="lazy-image-placeholder" style="width: 13.079ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad78382c3d23bcb4051b3148f1a23b1d0ba52e3" data-alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  arises from a unique m-by-n matrix A: explicitly, the (i, j)-entry of A is the ith coordinate of f (ej), where ej = (0, ..., 0, 1, 0, ..., 0) is the <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> with 1 in the jth position and 0 elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f. For example, the 2×2 matrix <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{bmatrix}a&c\\b&d\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}a&c\\b&d\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0996c87b9d5163ae1a5e132fd08eb0abced82042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.093ex; height:6.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}a&c\\b&d\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 13.093ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0996c87b9d5163ae1a5e132fd08eb0abced82042" data-alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}a&c\\b&d\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> can be viewed as the transform of the <a href="/wiki/Unit_square" title="Unit square">unit square</a> into a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d). The parallelogram pictured at the right is obtained by multiplying A with each of the column vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}},{\begin{bmatrix}1\\0\end{bmatrix}},{\begin{bmatrix}1\\1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}},{\begin{bmatrix}1\\0\end{bmatrix}},{\begin{bmatrix}1\\1\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee78c2dd87473db52d51e244cefb9172eea1bb58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.175ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}},{\begin{bmatrix}1\\0\end{bmatrix}},{\begin{bmatrix}1\\1\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 15.175ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee78c2dd87473db52d51e244cefb9172eea1bb58" data-alt="{\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}},{\begin{bmatrix}1\\0\end{bmatrix}},{\begin{bmatrix}1\\1\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0\\1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0\\1\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b48db099856bd05394c31adef4678aa9a6f9bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.369ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0\\1\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 4.369ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b48db099856bd05394c31adef4678aa9a6f9bee" data-alt="{\displaystyle {\begin{bmatrix}0\\1\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in turn. These vectors define the vertices of the unit square. The following table shows several 2×2 real matrices with the associated linear maps of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066b155c535a38739cc0c4b288324cbb7a4a227a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}.}"></noscript><span class="lazy-image-placeholder" style="width: 3.379ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066b155c535a38739cc0c4b288324cbb7a4a227a" data-alt="{\displaystyle \mathbb {R} ^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠ The blue original is mapped to the green grid and shapes. The origin (0, 0) is marked with a black point. <table class="wikitable" style="text-align:center; margin:1em auto 1em auto;"> <tbody><tr> <td><a href="/wiki/Shear_mapping" title="Shear mapping">Horizontal shear</a> with m = 1.25. </td> <td><a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">Reflection</a> through the vertical axis </td> <td><a href="/wiki/Squeeze_mapping" title="Squeeze mapping">Squeeze mapping</a> with r = 3/2 </td> <td><a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">Scaling</a> by a factor of 3/2 </td> <td><a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation</a> by π/6 = 30° </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&1.25\\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1.25</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&1.25\\0&1\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952686ab43a12dcc73b5a2faaef995e28c54af2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.826ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&1.25\\0&1\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 10.826ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952686ab43a12dcc73b5a2faaef995e28c54af2b" data-alt="{\displaystyle {\begin{bmatrix}1&1.25\\0&1\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}-1&0\\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}-1&0\\0&1\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ddc64d626699bfb8b3b1722d8156c6d71b3101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.662ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}-1&0\\0&1\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 9.662ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ddc64d626699bfb8b3b1722d8156c6d71b3101" data-alt="{\displaystyle {\begin{bmatrix}-1&0\\0&1\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {2}{3}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {2}{3}}\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed82dee99393f43ca58ca109a07882c8502039fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:9.101ex; height:7.843ex;" alt="{\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {2}{3}}\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 9.101ex;height: 7.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed82dee99393f43ca58ca109a07882c8502039fa" data-alt="{\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {2}{3}}\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {3}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {3}{2}}\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb4cd5330d9744daf8cce4670f5d7baeb8405c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:9.101ex; height:7.843ex;" alt="{\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {3}{2}}\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 9.101ex;height: 7.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb4cd5330d9744daf8cce4670f5d7baeb8405c1" data-alt="{\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {3}{2}}\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\cos \left({\frac {\pi }{6}}\right)&-\sin \left({\frac {\pi }{6}}\right)\\\sin \left({\frac {\pi }{6}}\right)&\cos \left({\frac {\pi }{6}}\right)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cos \left({\frac {\pi }{6}}\right)&-\sin \left({\frac {\pi }{6}}\right)\\\sin \left({\frac {\pi }{6}}\right)&\cos \left({\frac {\pi }{6}}\right)\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30963a4382db630e21570eedf5fa461cbf06cfb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:23.444ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}\cos \left({\frac {\pi }{6}}\right)&-\sin \left({\frac {\pi }{6}}\right)\\\sin \left({\frac {\pi }{6}}\right)&\cos \left({\frac {\pi }{6}}\right)\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 23.444ex;height: 9.843ex;vertical-align: -4.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30963a4382db630e21570eedf5fa461cbf06cfb7" data-alt="{\displaystyle {\begin{bmatrix}\cos \left({\frac {\pi }{6}}\right)&-\sin \left({\frac {\pi }{6}}\right)\\\sin \left({\frac {\pi }{6}}\right)&\cos \left({\frac {\pi }{6}}\right)\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td width="20%"><a href="/wiki/File:VerticalShear_m%3D1.25.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/VerticalShear_m%3D1.25.svg/175px-VerticalShear_m%3D1.25.svg.png" decoding="async" width="175" height="79" class="mw-file-element" data-file-width="1225" data-file-height="550"></noscript><span class="lazy-image-placeholder" style="width: 175px;height: 79px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/VerticalShear_m%3D1.25.svg/175px-VerticalShear_m%3D1.25.svg.png" data-width="175" data-height="79" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/VerticalShear_m%3D1.25.svg/263px-VerticalShear_m%3D1.25.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/VerticalShear_m%3D1.25.svg/350px-VerticalShear_m%3D1.25.svg.png 2x" data-class="mw-file-element"> </a> </td> <td width="20%"><a href="/wiki/File:Flip_map.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Flip_map.svg/150px-Flip_map.svg.png" decoding="async" width="150" height="97" class="mw-file-element" data-file-width="858" data-file-height="554"></noscript><span class="lazy-image-placeholder" style="width: 150px;height: 97px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Flip_map.svg/150px-Flip_map.svg.png" data-width="150" data-height="97" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Flip_map.svg/225px-Flip_map.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Flip_map.svg/300px-Flip_map.svg.png 2x" data-class="mw-file-element"> </a> </td> <td width="20%"><a href="/wiki/File:Squeeze_r%3D1.5.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Squeeze_r%3D1.5.svg/150px-Squeeze_r%3D1.5.svg.png" decoding="async" width="150" height="101" class="mw-file-element" data-file-width="820" data-file-height="550"></noscript><span class="lazy-image-placeholder" style="width: 150px;height: 101px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Squeeze_r%3D1.5.svg/150px-Squeeze_r%3D1.5.svg.png" data-width="150" data-height="101" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Squeeze_r%3D1.5.svg/225px-Squeeze_r%3D1.5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Squeeze_r%3D1.5.svg/300px-Squeeze_r%3D1.5.svg.png 2x" data-class="mw-file-element"> </a> </td> <td width="20%"><a href="/wiki/File:Scaling_by_1.5.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Scaling_by_1.5.svg/125px-Scaling_by_1.5.svg.png" decoding="async" width="125" height="125" class="mw-file-element" data-file-width="825" data-file-height="825"></noscript><span class="lazy-image-placeholder" style="width: 125px;height: 125px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Scaling_by_1.5.svg/125px-Scaling_by_1.5.svg.png" data-width="125" data-height="125" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Scaling_by_1.5.svg/188px-Scaling_by_1.5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Scaling_by_1.5.svg/250px-Scaling_by_1.5.svg.png 2x" data-class="mw-file-element"> </a> </td> <td width="20%"><a href="/wiki/File:Rotation_by_pi_over_6.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Rotation_by_pi_over_6.svg/125px-Rotation_by_pi_over_6.svg.png" decoding="async" width="125" height="125" class="mw-file-element" data-file-width="748" data-file-height="748"></noscript><span class="lazy-image-placeholder" style="width: 125px;height: 125px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Rotation_by_pi_over_6.svg/125px-Rotation_by_pi_over_6.svg.png" data-width="125" data-height="125" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Rotation_by_pi_over_6.svg/188px-Rotation_by_pi_over_6.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Rotation_by_pi_over_6.svg/250px-Rotation_by_pi_over_6.svg.png 2x" data-class="mw-file-element"> </a> </td></tr></tbody></table> Under the <a href="/wiki/Bijection" title="Bijection">1-to-1 correspondence</a> between matrices and linear maps, matrix multiplication corresponds to <a href="/wiki/Function_composition" title="Function composition">composition</a> of maps:<a href="#cite_note-23">[23]</a> if a k-by-m matrix B represents another linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {R} ^{m}\to \mathbb {R} ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} ^{m}\to \mathbb {R} ^{k}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc4b5d55e98cf86ea55aa1a57e470fc6b3076fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.787ex; height:3.009ex;" alt="{\displaystyle g:\mathbb {R} ^{m}\to \mathbb {R} ^{k}}"></noscript><span class="lazy-image-placeholder" style="width: 12.787ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc4b5d55e98cf86ea55aa1a57e470fc6b3076fe" data-alt="{\displaystyle g:\mathbb {R} ^{m}\to \mathbb {R} ^{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , then the composition g ∘ f is represented by BA since <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g\circ f)({\mathbf {x}})=g(f({\mathbf {x}}))=g({\mathbf {Ax}})={\mathbf {B}}({\mathbf {Ax}})=({\mathbf {BA}}){\mathbf {x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∘</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\circ f)({\mathbf {x}})=g(f({\mathbf {x}}))=g({\mathbf {Ax}})={\mathbf {B}}({\mathbf {Ax}})=({\mathbf {BA}}){\mathbf {x}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7449df2e95a0fe52fd087e3f8d3b32e09606c338" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.721ex; height:2.843ex;" alt="{\displaystyle (g\circ f)({\mathbf {x}})=g(f({\mathbf {x}}))=g({\mathbf {Ax}})={\mathbf {B}}({\mathbf {Ax}})=({\mathbf {BA}}){\mathbf {x}}.}"></noscript><span class="lazy-image-placeholder" style="width: 50.721ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7449df2e95a0fe52fd087e3f8d3b32e09606c338" data-alt="{\displaystyle (g\circ f)({\mathbf {x}})=g(f({\mathbf {x}}))=g({\mathbf {Ax}})={\mathbf {B}}({\mathbf {Ax}})=({\mathbf {BA}}){\mathbf {x}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The last equality follows from the above-mentioned associativity of matrix multiplication. The <a href="/wiki/Rank_of_a_matrix" class="mw-redirect" title="Rank of a matrix">rank of a matrix</a> A is the maximum number of <a href="/wiki/Linear_independence" title="Linear independence">linearly independent</a> row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors.<a href="#cite_note-24">[24]</a> Equivalently it is the <a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">dimension</a> of the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of the linear map represented by A.<a href="#cite_note-25">[25]</a> The <a href="/wiki/Rank%E2%80%93nullity_theorem" title="Rank–nullity theorem">rank–nullity theorem</a> states that the dimension of the <a href="/wiki/Kernel_(matrix)" class="mw-redirect" title="Kernel (matrix)">kernel</a> of a matrix plus the rank equals the number of columns of the matrix.<a href="#cite_note-26">[26]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Square_matrix">Square matrix</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=11" title="Edit section: Square matrix" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Square_matrix" title="Square matrix">Square matrix</a></div> A <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> is a matrix with the same number of rows and columns.<a href="#cite_note-:4-5">[5]</a> An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries aii form the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix. <div class="mw-heading mw-heading3"><h3 id="Main_types">Main types</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=12" title="Edit section: Main types" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <dl><dd><table class="wikitable" style="float:right; margin:0ex 0ex 2ex 2ex;"> <tbody><tr> <th>Name</th> <th>Example with n = 3 </th></tr> <tr> <td><a href="/wiki/Diagonal_matrix" title="Diagonal matrix">Diagonal matrix</a></td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\\\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c1cbd586cf9fd90a642b4a7ca5e78e92418557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.815ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\\\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 17.815ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c1cbd586cf9fd90a642b4a7ca5e78e92418557" data-alt="{\displaystyle {\begin{bmatrix}a_{11}&0&0\\0&a_{22}&0\\0&0&a_{33}\\\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/Lower_triangular_matrix" class="mw-redirect" title="Lower triangular matrix">Lower triangular matrix</a></td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0cfe35165701bd0692e6063e5fca0c636b5b905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.815ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 17.815ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0cfe35165701bd0692e6063e5fca0c636b5b905" data-alt="{\displaystyle {\begin{bmatrix}a_{11}&0&0\\a_{21}&a_{22}&0\\a_{31}&a_{32}&a_{33}\\\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/Upper_triangular_matrix" class="mw-redirect" title="Upper triangular matrix">Upper triangular matrix</a></td> <td style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\\\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a33db4eee227aa187ac0a47dfbe1079336bcae86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.815ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\\\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 17.815ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a33db4eee227aa187ac0a47dfbe1079336bcae86" data-alt="{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\0&a_{22}&a_{23}\\0&0&a_{33}\\\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Diagonal_and_triangular_matrix">Diagonal and triangular matrix</h4> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=13" title="Edit section: Diagonal and triangular matrix" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> If all entries of A below the main diagonal are zero, A is called an upper <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular matrix</a>. Similarly, if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries outside the main diagonal are zero, A is called a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a>. <div class="mw-heading mw-heading4"><h4 id="Identity_matrix">Identity matrix</h4> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=14" title="Edit section: Identity matrix" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 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/Identity_matrix" title="Identity matrix">Identity matrix</a></div> The identity matrix In of size n is the n-by-n matrix in which all the elements on the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> are equal to 1 and all other elements are equal to 0, for example, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {I} _{1}&={\begin{bmatrix}1\end{bmatrix}},\\[4pt]\mathbf {I} _{2}&={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\\[4pt]\vdots &\\[4pt]\mathbf {I} _{n}&={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\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="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {I} _{1}&={\begin{bmatrix}1\end{bmatrix}},\\[4pt]\mathbf {I} _{2}&={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\\[4pt]\vdots &\\[4pt]\mathbf {I} _{n}&={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f691509cf9deb5416f60f917b8dfc543b1c6d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.671ex; width:23.368ex; height:30.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {I} _{1}&={\begin{bmatrix}1\end{bmatrix}},\\[4pt]\mathbf {I} _{2}&={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\\[4pt]\vdots &\\[4pt]\mathbf {I} _{n}&={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 23.368ex;height: 30.509ex;vertical-align: -14.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f691509cf9deb5416f60f917b8dfc543b1c6d3" data-alt="{\displaystyle {\begin{aligned}\mathbf {I} _{1}&={\begin{bmatrix}1\end{bmatrix}},\\[4pt]\mathbf {I} _{2}&={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\\[4pt]\vdots &\\[4pt]\mathbf {I} _{n}&={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  It is a square matrix of order n, and also a special kind of <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a>. It is called an identity matrix because multiplication with it leaves a matrix unchanged: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {AI}}_{n}={\mathbf {I}}_{m}{\mathbf {A}}={\mathbf {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {AI}}_{n}={\mathbf {I}}_{m}{\mathbf {A}}={\mathbf {A}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f23e8b30561da7e7566d0b4b6f6e2e3ea424d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.177ex; height:2.509ex;" alt="{\displaystyle {\mathbf {AI}}_{n}={\mathbf {I}}_{m}{\mathbf {A}}={\mathbf {A}}}"></noscript><span class="lazy-image-placeholder" style="width: 17.177ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f23e8b30561da7e7566d0b4b6f6e2e3ea424d6" data-alt="{\displaystyle {\mathbf {AI}}_{n}={\mathbf {I}}_{m}{\mathbf {A}}={\mathbf {A}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  for any m-by-n matrix A. A nonzero scalar multiple of an identity matrix is called a scalar matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field. <div class="mw-heading mw-heading4"><h4 id="Symmetric_or_skew-symmetric_matrix">Symmetric or skew-symmetric matrix</h4> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=15" title="Edit section: Symmetric or skew-symmetric matrix" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> A square matrix A that is equal to its transpose, that is, A = AT, is a <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a>. If instead, A is equal to the negative of its transpose, that is, A = −AT, then A is a <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric matrix</a>. In complex matrices, symmetry is often replaced by the concept of <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrices</a>, which satisfies A∗ = A, where the star or <a href="/wiki/Asterisk" title="Asterisk">asterisk</a> denotes the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of the matrix, that is, the transpose of the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of A. By the <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a>, real symmetric matrices and complex Hermitian matrices have an <a href="/wiki/Eigenbasis" class="mw-redirect" title="Eigenbasis">eigenbasis</a>; that is, every vector is expressible as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of eigenvectors. In both cases, all eigenvalues are real.<a href="#cite_note-27">[27]</a> This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see <a href="#Infinite_matrices">below</a>. <div class="mw-heading mw-heading4"><h4 id="Invertible_matrix_and_its_inverse">Invertible matrix and its inverse</h4> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=16" title="Edit section: Invertible matrix and its inverse" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> A square matrix A is called <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> or non-singular if there exists a matrix B such that<a href="#cite_note-28">[28]</a><a href="#cite_note-29">[29]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {AB}}={\mathbf {BA}}={\mathbf {I}}_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {AB}}={\mathbf {BA}}={\mathbf {I}}_{n},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3183e5c83ae58ed74e6dea8bf59174800f86f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.917ex; height:2.509ex;" alt="{\displaystyle {\mathbf {AB}}={\mathbf {BA}}={\mathbf {I}}_{n},}"></noscript><span class="lazy-image-placeholder" style="width: 16.917ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3183e5c83ae58ed74e6dea8bf59174800f86f2" data-alt="{\displaystyle {\mathbf {AB}}={\mathbf {BA}}={\mathbf {I}}_{n},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where In is the n×n <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> with 1s on the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> and 0s elsewhere. If B exists, it is unique and is called the <a href="/wiki/Invertible_matrix" title="Invertible matrix">inverse matrix</a> of A, denoted A−1. <div class="mw-heading mw-heading4"><h4 id="Definite_matrix">Definite matrix</h4> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=17" title="Edit section: Definite matrix" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <table class="wikitable" style="float:right; text-align:center; margin:0ex 0ex 2ex 2ex;"> <tbody><tr> <th><a href="/wiki/Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">Positive definite matrix</a></th> <th><a href="/wiki/Indefinite_matrix" class="mw-redirect" title="Indefinite matrix">Indefinite matrix</a> </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&1\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&1\\\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6b3f39260b6d0dee463e5765d97348abe2433c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.605ex; height:7.509ex;" alt="{\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&1\\\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 8.605ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6b3f39260b6d0dee463e5765d97348abe2433c" data-alt="{\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&1\\\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&-{\frac {1}{4}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&-{\frac {1}{4}}\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdcf7cd95e30f375ac7f0bf077507688e78a758a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.909ex; height:7.509ex;" alt="{\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&-{\frac {1}{4}}\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 10.909ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdcf7cd95e30f375ac7f0bf077507688e78a758a" data-alt="{\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&-{\frac {1}{4}}\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x,y)={\frac {1}{4}}x^{2}+y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x,y)={\frac {1}{4}}x^{2}+y^{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da7a2535f99b29a4deb6ec7f4e3e1d3494f6e12d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.703ex; height:5.176ex;" alt="{\displaystyle Q(x,y)={\frac {1}{4}}x^{2}+y^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 19.703ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da7a2535f99b29a4deb6ec7f4e3e1d3494f6e12d" data-alt="{\displaystyle Q(x,y)={\frac {1}{4}}x^{2}+y^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x,y)={\frac {1}{4}}x^{2}-{\frac {1}{4}}y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x,y)={\frac {1}{4}}x^{2}-{\frac {1}{4}}y^{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8556380d0b0d296c6ac1b3f56b5730429eb04113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.701ex; height:5.176ex;" alt="{\displaystyle Q(x,y)={\frac {1}{4}}x^{2}-{\frac {1}{4}}y^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 21.701ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8556380d0b0d296c6ac1b3f56b5730429eb04113" data-alt="{\displaystyle Q(x,y)={\frac {1}{4}}x^{2}-{\frac {1}{4}}y^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td><a href="/wiki/File:Ellipse_in_coordinate_system_with_semi-axes_labelled.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Ellipse_in_coordinate_system_with_semi-axes_labelled.svg/150px-Ellipse_in_coordinate_system_with_semi-axes_labelled.svg.png" decoding="async" width="150" height="100" class="mw-file-element" data-file-width="512" data-file-height="341"></noscript><span class="lazy-image-placeholder" style="width: 150px;height: 100px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Ellipse_in_coordinate_system_with_semi-axes_labelled.svg/150px-Ellipse_in_coordinate_system_with_semi-axes_labelled.svg.png" data-width="150" data-height="100" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Ellipse_in_coordinate_system_with_semi-axes_labelled.svg/225px-Ellipse_in_coordinate_system_with_semi-axes_labelled.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Ellipse_in_coordinate_system_with_semi-axes_labelled.svg/300px-Ellipse_in_coordinate_system_with_semi-axes_labelled.svg.png 2x" data-class="mw-file-element"> </a> Points such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Q(x,y)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Q(x,y)=1}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a7b404d61be48d4968c117c8cefaf259e838dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.428ex; height:2.843ex;" alt="{\textstyle Q(x,y)=1}"></noscript><span class="lazy-image-placeholder" style="width: 11.428ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a7b404d61be48d4968c117c8cefaf259e838dd" data-alt="{\textstyle Q(x,y)=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (<a href="/wiki/Ellipse" title="Ellipse">Ellipse</a>) </td> <td><a href="/wiki/File:Hyperbola2_SVG.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Hyperbola2_SVG.svg/150px-Hyperbola2_SVG.svg.png" decoding="async" width="150" height="150" class="mw-file-element" data-file-width="512" data-file-height="512"></noscript><span class="lazy-image-placeholder" style="width: 150px;height: 150px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Hyperbola2_SVG.svg/150px-Hyperbola2_SVG.svg.png" data-width="150" data-height="150" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Hyperbola2_SVG.svg/225px-Hyperbola2_SVG.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Hyperbola2_SVG.svg/300px-Hyperbola2_SVG.svg.png 2x" data-class="mw-file-element"> </a> Points such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Q(x,y)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Q(x,y)=1}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a7b404d61be48d4968c117c8cefaf259e838dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.428ex; height:2.843ex;" alt="{\textstyle Q(x,y)=1}"></noscript><span class="lazy-image-placeholder" style="width: 11.428ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73a7b404d61be48d4968c117c8cefaf259e838dd" data-alt="{\textstyle Q(x,y)=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (<a href="/wiki/Hyperbola" title="Hyperbola">Hyperbola</a>) </td></tr></tbody></table> A symmetric real matrix A is called <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite</a> if the associated <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\mathbf {x}})={\mathbf {x}}^{\rm {T}}{\mathbf {Ax}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f({\mathbf {x}})={\mathbf {x}}^{\rm {T}}{\mathbf {Ax}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2424901a8cf7b01fee4adbba41b4497930367b08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.858ex; height:3.176ex;" alt="{\displaystyle f({\mathbf {x}})={\mathbf {x}}^{\rm {T}}{\mathbf {Ax}}}"></noscript><span class="lazy-image-placeholder" style="width: 13.858ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2424901a8cf7b01fee4adbba41b4497930367b08" data-alt="{\displaystyle f({\mathbf {x}})={\mathbf {x}}^{\rm {T}}{\mathbf {Ax}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  has a positive value for every nonzero vector x in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ef548febfc9981762740107858be9e3a5576c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}.}"></noscript><span class="lazy-image-placeholder" style="width: 3.543ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ef548febfc9981762740107858be9e3a5576c3" data-alt="{\displaystyle \mathbb {R} ^{n}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠ If f (x) only yields negative values then A is <a href="/wiki/Definiteness_of_a_matrix#Negative_definite" class="mw-redirect" title="Definiteness of a matrix">negative-definite</a>; if f does produce both negative and positive values then A is <a href="/wiki/Definiteness_of_a_matrix#Indefinite" class="mw-redirect" title="Definiteness of a matrix"> indefinite</a>.<a href="#cite_note-30">[30]</a> If the quadratic form f yields only non-negative values (positive or zero), the symmetric matrix is called positive-semidefinite (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible.<a href="#cite_note-31">[31]</a> The table at the right shows two possibilities for 2-by-2 matrices. Allowing as input two different vectors instead yields the <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a> associated to A:<a href="#cite_note-32">[32]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\mathbf {A}}({\mathbf {x}},{\mathbf {y}})={\mathbf {x}}^{\rm {T}}{\mathbf {Ay}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">y</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{\mathbf {A}}({\mathbf {x}},{\mathbf {y}})={\mathbf {x}}^{\rm {T}}{\mathbf {Ay}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a051ec59e1d36edd416b57f306d96b54811d40e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.095ex; height:3.176ex;" alt="{\displaystyle B_{\mathbf {A}}({\mathbf {x}},{\mathbf {y}})={\mathbf {x}}^{\rm {T}}{\mathbf {Ay}}.}"></noscript><span class="lazy-image-placeholder" style="width: 19.095ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a051ec59e1d36edd416b57f306d96b54811d40e" data-alt="{\displaystyle B_{\mathbf {A}}({\mathbf {x}},{\mathbf {y}})={\mathbf {x}}^{\rm {T}}{\mathbf {Ay}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  In the case of complex matrices, the same terminology and result apply, with symmetric matrix, quadratic form, bilinear form, and transpose xT replaced respectively by <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrix</a>, <a href="/wiki/Hermitian_form" class="mw-redirect" title="Hermitian form">Hermitian form</a>, <a href="/wiki/Sesquilinear_form" title="Sesquilinear form">sesquilinear form</a>, and <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> xH. <div class="mw-heading mw-heading4"><h4 id="Orthogonal_matrix">Orthogonal matrix</h4> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=18" title="Edit section: Orthogonal matrix" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 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/Orthogonal_matrix" title="Orthogonal matrix">Orthogonal matrix</a></div> An orthogonal matrix is a square matrix with <a href="/wiki/Real_number" title="Real number">real</a> entries whose columns and rows are <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> <a href="/wiki/Unit_vector" title="Unit vector">unit vectors</a> (that is, <a href="/wiki/Orthonormality" title="Orthonormality">orthonormal</a> vectors). Equivalently, a matrix A is orthogonal if its <a href="/wiki/Transpose" title="Transpose">transpose</a> is equal to its <a href="/wiki/Invertible_matrix" title="Invertible matrix">inverse</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ^{\mathrm {T} }=\mathbf {A} ^{-1},\,}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ^{\mathrm {T} }=\mathbf {A} ^{-1},\,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f07fbcf3f405d35734fe7d362ec752cec6582b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.923ex; height:3.009ex;" alt="{\displaystyle \mathbf {A} ^{\mathrm {T} }=\mathbf {A} ^{-1},\,}"></noscript><span class="lazy-image-placeholder" style="width: 11.923ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f07fbcf3f405d35734fe7d362ec752cec6582b" data-alt="{\displaystyle \mathbf {A} ^{\mathrm {T} }=\mathbf {A} ^{-1},\,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> which entails <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ^{\mathrm {T} }\mathbf {A} =\mathbf {A} \mathbf {A} ^{\mathrm {T} }=\mathbf {I} _{n},}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ^{\mathrm {T} }\mathbf {A} =\mathbf {A} \mathbf {A} ^{\mathrm {T} }=\mathbf {I} _{n},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e78d5149a8e1b1ad0708fa65a56aa43879587d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.992ex; height:3.009ex;" alt="{\displaystyle \mathbf {A} ^{\mathrm {T} }\mathbf {A} =\mathbf {A} \mathbf {A} ^{\mathrm {T} }=\mathbf {I} _{n},}"></noscript><span class="lazy-image-placeholder" style="width: 19.992ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e78d5149a8e1b1ad0708fa65a56aa43879587d0" data-alt="{\displaystyle \mathbf {A} ^{\mathrm {T} }\mathbf {A} =\mathbf {A} \mathbf {A} ^{\mathrm {T} }=\mathbf {I} _{n},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> where In is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> of size n. An orthogonal matrix A is necessarily <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> (with inverse A−1 = AT), <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary</a> (A−1 = A*), and <a href="/wiki/Normal_matrix" title="Normal matrix">normal</a> (A*A = AA*). The <a href="/wiki/Determinant" title="Determinant">determinant</a> of any orthogonal matrix is either +1 or −1. A special orthogonal matrix is an orthogonal matrix with <a href="/wiki/Determinant" title="Determinant">determinant</a> +1. As a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a>, every orthogonal matrix with determinant +1 is a pure <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a> without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant -1 reverses the orientation, i.e., is a composition of a pure <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> and a (possibly null) rotation. The identity matrices have determinant 1 and are pure rotations by an angle zero. The <a href="/wiki/Complex_number" title="Complex number">complex</a> analog of an orthogonal matrix is a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>. <div class="mw-heading mw-heading3"><h3 id="Main_operations">Main operations</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=19" title="Edit section: Main operations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <div class="mw-heading mw-heading4"><h4 id="Trace">Trace</h4> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=20" title="Edit section: Trace" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The <a href="/wiki/Trace_of_a_matrix" class="mw-redirect" title="Trace of a matrix">trace</a>, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned <a href="#_noncommutative">above</a>, the trace of the product of two matrices is independent of the order of the factors: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {AB} )=\operatorname {tr} (\mathbf {BA} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <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">B</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {AB} )=\operatorname {tr} (\mathbf {BA} ).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348b05db9499e85326f9e41f5bcb6cf7eb053350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.837ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {AB} )=\operatorname {tr} (\mathbf {BA} ).}"></noscript><span class="lazy-image-placeholder" style="width: 18.837ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348b05db9499e85326f9e41f5bcb6cf7eb053350" data-alt="{\displaystyle \operatorname {tr} (\mathbf {AB} )=\operatorname {tr} (\mathbf {BA} ).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> This is immediate from the definition of matrix multiplication: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {AB} )=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\operatorname {tr} (\mathbf {BA} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {AB} )=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\operatorname {tr} (\mathbf {BA} ).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0caa4f820bb74d566e181ba5cf3d09181a2e9cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.602ex; height:7.176ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {AB} )=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\operatorname {tr} (\mathbf {BA} ).}"></noscript><span class="lazy-image-placeholder" style="width: 34.602ex;height: 7.176ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0caa4f820bb74d566e181ba5cf3d09181a2e9cb8" data-alt="{\displaystyle \operatorname {tr} (\mathbf {AB} )=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\operatorname {tr} (\mathbf {BA} ).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> It follows that the trace of the product of more than two matrices is independent of <a href="/wiki/Cyclic_permutation" title="Cyclic permutation">cyclic permutations</a> of the matrices, however, this does not in general apply for arbitrary permutations (for example, tr(ABC) ≠ tr(BAC), in general). Also, the trace of a matrix is equal to that of its transpose, that is, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} ({\mathbf {A}})=\operatorname {tr} ({\mathbf {A}}^{\rm {T}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} ({\mathbf {A}})=\operatorname {tr} ({\mathbf {A}}^{\rm {T}}).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06c85c23ef3930b8eb98802873e47d727ad1ec49" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.454ex; height:3.176ex;" alt="{\displaystyle \operatorname {tr} ({\mathbf {A}})=\operatorname {tr} ({\mathbf {A}}^{\rm {T}}).}"></noscript><span class="lazy-image-placeholder" style="width: 16.454ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06c85c23ef3930b8eb98802873e47d727ad1ec49" data-alt="{\displaystyle \operatorname {tr} ({\mathbf {A}})=\operatorname {tr} ({\mathbf {A}}^{\rm {T}}).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <div class="mw-heading mw-heading4"><h4 id="Determinant">Determinant</h4> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=21" title="Edit section: Determinant" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 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/Determinant" title="Determinant">Determinant</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Determinant_example.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Determinant_example.svg/300px-Determinant_example.svg.png" decoding="async" width="300" height="126" class="mw-file-element" data-file-width="512" data-file-height="215"></noscript><span class="lazy-image-placeholder" style="width: 300px;height: 126px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Determinant_example.svg/300px-Determinant_example.svg.png" data-width="300" data-height="126" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Determinant_example.svg/450px-Determinant_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Determinant_example.svg/600px-Determinant_example.svg.png 2x" data-class="mw-file-element"> </a><figcaption>A linear transformation on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.732ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" data-alt="{\displaystyle \mathbb {R} ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠ given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the <a href="/wiki/Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a>, since it turns the counterclockwise orientation of the vectors to a clockwise one.</figcaption></figure> The determinant of a square matrix A (denoted det(A) or |A|) is a number encoding certain properties of the matrix. A matrix is invertible <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> its determinant is nonzero. Its <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> equals the area (in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.732ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" data-alt="{\displaystyle \mathbb {R} ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠) or volume (in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></noscript><span class="lazy-image-placeholder" style="width: 2.732ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" data-alt="{\displaystyle \mathbb {R} ^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved. The determinant of 2-by-2 matrices is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\begin{bmatrix}a&b\\c&d\end{bmatrix}}=ad-bc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\begin{bmatrix}a&b\\c&d\end{bmatrix}}=ad-bc.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8138b2141d7b9406cebd3eb82b3cf05b238ca851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.627ex; height:6.176ex;" alt="{\displaystyle \det {\begin{bmatrix}a&b\\c&d\end{bmatrix}}=ad-bc.}"></noscript><span class="lazy-image-placeholder" style="width: 22.627ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8138b2141d7b9406cebd3eb82b3cf05b238ca851" data-alt="{\displaystyle \det {\begin{bmatrix}a&b\\c&d\end{bmatrix}}=ad-bc.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> <a href="#cite_note-:3-33">[33]</a></dd></dl> The determinant of 3-by-3 matrices involves 6 terms (<a href="/wiki/Rule_of_Sarrus" title="Rule of Sarrus">rule of Sarrus</a>). The more lengthy <a href="/wiki/Leibniz_formula_for_determinants" title="Leibniz formula for determinants">Leibniz formula</a> generalizes these two formulae to all dimensions.<a href="#cite_note-34">[34]</a> The determinant of a product of square matrices equals the product of their determinants: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det({\mathbf {AB}})=\det({\mathbf {A}})\cdot \det({\mathbf {B}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det({\mathbf {AB}})=\det({\mathbf {A}})\cdot \det({\mathbf {B}}),}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41efa7afe8f21c3ccbd03f26ae04e75dd8970157" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.382ex; height:2.843ex;" alt="{\displaystyle \det({\mathbf {AB}})=\det({\mathbf {A}})\cdot \det({\mathbf {B}}),}"></noscript><span class="lazy-image-placeholder" style="width: 28.382ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41efa7afe8f21c3ccbd03f26ae04e75dd8970157" data-alt="{\displaystyle \det({\mathbf {AB}})=\det({\mathbf {A}})\cdot \det({\mathbf {B}}),}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  or using alternate notation:<a href="#cite_note-35">[35]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\mathbf {AB}}|=|{\mathbf {A}}|\cdot |{\mathbf {B}}|.}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\mathbf {AB}}|=|{\mathbf {A}}|\cdot |{\mathbf {B}}|.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c1367abab8f34a3bcea0c4eb1c3751555af3c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.147ex; height:2.843ex;" alt="{\displaystyle |{\mathbf {AB}}|=|{\mathbf {A}}|\cdot |{\mathbf {B}}|.}"></noscript><span class="lazy-image-placeholder" style="width: 17.147ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c1367abab8f34a3bcea0c4eb1c3751555af3c1" data-alt="{\displaystyle |{\mathbf {AB}}|=|{\mathbf {A}}|\cdot |{\mathbf {B}}|.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Adding a multiple of any row to another row, or a multiple of any column to another column does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1.<a href="#cite_note-36">[36]</a> Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices, the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the <a href="/wiki/Laplace_expansion" title="Laplace expansion">Laplace expansion</a> expresses the determinant in terms of <a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">minors</a>, that is, determinants of smaller matrices.<a href="#cite_note-37">[37]</a> This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve <a href="/wiki/Linear_system" title="Linear system">linear systems</a> using <a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a>, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.<a href="#cite_note-38">[38]</a> <div class="mw-heading mw-heading4"><h4 id="Eigenvalues_and_eigenvectors">Eigenvalues and eigenvectors</h4> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=22" title="Edit section: Eigenvalues and eigenvectors" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 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/Eigenvalue,_eigenvector_and_eigenspace" class="mw-redirect" title="Eigenvalue, eigenvector and eigenspace">Eigenvalues and eigenvectors</a></div> A number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lambda }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f801b46dadd4b4d0ba4c6592b232053bd79e080b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\textstyle \lambda }"></noscript><span class="lazy-image-placeholder" style="width: 1.355ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f801b46dadd4b4d0ba4c6592b232053bd79e080b" data-alt="{\textstyle \lambda }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and a non-zero vector v satisfying <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \mathbf {v} =\lambda \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"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \mathbf {v} =\lambda \mathbf {v} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4bca75ce898b48e51ff4f79b0a8cc3c53afdee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.295ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} \mathbf {v} =\lambda \mathbf {v} }"></noscript><span class="lazy-image-placeholder" style="width: 9.295ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4bca75ce898b48e51ff4f79b0a8cc3c53afdee" data-alt="{\displaystyle \mathbf {A} \mathbf {v} =\lambda \mathbf {v} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> are called an eigenvalue and an eigenvector of A, respectively.<a href="#cite_note-39">[39]</a><a href="#cite_note-40">[40]</a> The number λ is an eigenvalue of an n×n-matrix A if and only if (A − λIn) is not invertible, which is <a href="/wiki/Logical_equivalence" title="Logical equivalence">equivalent</a> to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\mathbf {A} -\lambda \mathbf {I} )=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(\mathbf {A} -\lambda \mathbf {I} )=0.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2df8eef1d708ca2a7cfbc4fa8fc10088ddc2b26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.176ex; height:2.843ex;" alt="{\displaystyle \det(\mathbf {A} -\lambda \mathbf {I} )=0.}"></noscript><span class="lazy-image-placeholder" style="width: 17.176ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2df8eef1d708ca2a7cfbc4fa8fc10088ddc2b26" data-alt="{\displaystyle \det(\mathbf {A} -\lambda \mathbf {I} )=0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> <a href="#cite_note-41">[41]</a></dd></dl> The polynomial pA in an <a href="/wiki/Indeterminate_(variable)" title="Indeterminate (variable)">indeterminate</a> X given by evaluation of the determinant det(X In − A) is called the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of A. It is a <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a> of <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> n. Therefore the polynomial equation pA(λ) = 0 has at most n different solutions, that is, eigenvalues of the matrix.<a href="#cite_note-42">[42]</a> They may be complex even if the entries of A are real. According to the <a href="/wiki/Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley–Hamilton theorem</a>, pA(A) = 0, that is, the result of substituting the matrix itself into its characteristic polynomial yields the <a href="/wiki/Zero_matrix" title="Zero matrix">zero matrix</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="Computational_aspects">Computational aspects</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=23" title="Edit section: Computational aspects" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms and iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a <a href="/wiki/Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequence</a> of vectors xn <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">converging</a> to an eigenvector when n tends to <a href="/wiki/Infinity" title="Infinity">infinity</a>.<a href="#cite_note-43">[43]</a> To choose the most appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms. The domain studying these matters is called <a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebra</a>.<a href="#cite_note-44">[44]</a> As with other numerical situations, two main aspects are the <a href="/wiki/Complexity_analysis" class="mw-redirect" title="Complexity analysis">complexity</a> of algorithms and their <a href="/wiki/Numerical_stability" title="Numerical stability">numerical stability</a>. Determining the complexity of an algorithm means finding <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bounds</a> or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, for example, <a href="/wiki/Matrix_multiplication_algorithm" title="Matrix multiplication algorithm">multiplication of matrices</a>. Calculating the matrix product of two n-by-n matrices using the definition given above needs n3 multiplications, since for any of the n2 entries of the product, n multiplications are necessary. The <a href="/wiki/Strassen_algorithm" title="Strassen algorithm">Strassen algorithm</a> outperforms this "naive" algorithm; it needs only n2.807 multiplications.<a href="#cite_note-45">[45]</a> A refined approach also incorporates specific features of the computing devices. In many practical situations, additional information about the matrices involved is known. An important case is <a href="/wiki/Sparse_matrix" title="Sparse matrix">sparse matrices</a>, that is, matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the <a href="/wiki/Conjugate_gradient_method" title="Conjugate gradient method">conjugate gradient method</a>.<a href="#cite_note-46">[46]</a> An algorithm is, roughly speaking, numerically stable if little deviations in the input values do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace expansion (adj(A) denotes the <a href="/wiki/Adjugate_matrix" title="Adjugate matrix">adjugate matrix</a> of A) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {A}}^{-1}=\operatorname {adj} ({\mathbf {A}})/\det({\mathbf {A}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A}}^{-1}=\operatorname {adj} ({\mathbf {A}})/\det({\mathbf {A}})}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b8cc0e88357362edc4bc9e8ad9df1559a4f310" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.054ex; height:3.176ex;" alt="{\displaystyle {\mathbf {A}}^{-1}=\operatorname {adj} ({\mathbf {A}})/\det({\mathbf {A}})}"></noscript><span class="lazy-image-placeholder" style="width: 23.054ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b8cc0e88357362edc4bc9e8ad9df1559a4f310" data-alt="{\displaystyle {\mathbf {A}}^{-1}=\operatorname {adj} ({\mathbf {A}})/\det({\mathbf {A}})}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  may lead to significant rounding errors if the determinant of the matrix is very small. The <a href="/wiki/Matrix_norm" title="Matrix norm">norm of a matrix</a> can be used to capture the <a href="/wiki/Condition_number" title="Condition number">conditioning</a> of linear algebraic problems, such as computing a matrix's inverse.<a href="#cite_note-47">[47]</a> Most computer <a href="/wiki/Programming_language" title="Programming language">programming languages</a> support arrays but are not designed with built-in commands for matrices. Instead, available external libraries provide matrix operations on arrays, in nearly all currently used programming languages. Matrix manipulation was among the earliest numerical applications of computers.<a href="#cite_note-48">[48]</a> The original <a href="/wiki/Dartmouth_BASIC" title="Dartmouth BASIC">Dartmouth BASIC</a> had built-in commands for matrix arithmetic on arrays from its <a href="/wiki/Dartmouth_BASIC#Second_Edition,_CARDBASIC" title="Dartmouth BASIC">second edition</a> implementation in 1964. As early as the 1970s, some engineering desktop computers such as the <a href="/wiki/HP_9830" class="mw-redirect" title="HP 9830">HP 9830</a> had <a href="/wiki/HP_9800_series#Programming" title="HP 9800 series">ROM cartridges to add BASIC commands for matrices</a>. Some computer languages such as <a href="/wiki/APL_(programming_language)" title="APL (programming language)">APL</a> were designed to manipulate matrices, and <a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">various mathematical programs</a> can be used to aid computing with matrices.<a href="#cite_note-49">[49]</a> As of 2023, most computers have some form of built-in matrix operations at a low level implementing the standard <a href="/wiki/BLAS" class="mw-redirect" title="BLAS">BLAS</a> specification, upon which most higher-level matrix and linear algebra libraries (e.g., <a href="/wiki/EISPACK" title="EISPACK">EISPACK</a>, <a href="/wiki/LINPACK" title="LINPACK">LINPACK</a>, <a href="/wiki/LAPACK" title="LAPACK">LAPACK</a>) rely. While most of these libraries require a professional level of coding, <a href="/wiki/LAPACK" title="LAPACK">LAPACK</a> can be accessed by higher-level (and user-friendly) bindings such as <a href="/wiki/NumPy" title="NumPy">NumPy</a>/<a href="/wiki/SciPy" title="SciPy">SciPy</a>, <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>, <a href="/wiki/GNU_Octave" title="GNU Octave">GNU Octave</a>, <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a>. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="Decomposition">Decomposition</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=24" title="Edit section: Decomposition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Matrix_decomposition" title="Matrix decomposition">Matrix decomposition</a>, <a href="/wiki/Matrix_diagonalization" class="mw-redirect" title="Matrix diagonalization">Matrix diagonalization</a>, <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>, and <a href="/wiki/Bareiss_algorithm" title="Bareiss algorithm">Bareiss algorithm</a></div> There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix decomposition or matrix factorization techniques. The interest of all these techniques is that they preserve certain properties of the matrices in question, such as determinant, rank, or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices. The <a href="/wiki/LU_decomposition" title="LU decomposition">LU decomposition</a> factors matrices as a product of lower (L) and an upper <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular matrices</a> (U).<a href="#cite_note-50">[50]</a> Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called <a href="/wiki/Forward_substitution" class="mw-redirect" title="Forward substitution">forward and back substitution</a>. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to <a href="/wiki/Row_echelon_form" title="Row echelon form">row echelon form</a>.<a href="#cite_note-51">[51]</a> Both methods proceed by multiplying the matrix by suitable <a href="/wiki/Elementary_matrix" title="Elementary matrix">elementary matrices</a>, which correspond to <a href="/wiki/Permutation_matrix" title="Permutation matrix">permuting rows or columns</a> and adding multiples of one row to another row. <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a> expresses any matrix A as a product UDV∗, where U and V are <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrices</a> and D is a diagonal matrix. <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Jordan_blocks.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Jordan_blocks.svg/250px-Jordan_blocks.svg.png" decoding="async" width="250" height="233" class="mw-file-element" data-file-width="180" data-file-height="168"></noscript><span class="lazy-image-placeholder" style="width: 250px;height: 233px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Jordan_blocks.svg/250px-Jordan_blocks.svg.png" data-width="250" data-height="233" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Jordan_blocks.svg/375px-Jordan_blocks.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Jordan_blocks.svg/500px-Jordan_blocks.svg.png 2x" data-class="mw-file-element"> </a><figcaption>An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks.</figcaption></figure> The <a href="/wiki/Eigendecomposition" class="mw-redirect" title="Eigendecomposition">eigendecomposition</a> or diagonalization expresses A as a product VDV−1, where D is a diagonal matrix and V is a suitable invertible matrix.<a href="#cite_note-52">[52]</a> If A can be written in this form, it is called <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a>. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into <a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a>, that is to say matrices whose only nonzero entries are the eigenvalues λ1 to λn of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right.<a href="#cite_note-53">[53]</a> Given the eigendecomposition, the nth power of A (that is, n-fold iterated matrix multiplication) can be calculated via <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {A}}^{n}=({\mathbf {VDV}}^{-1})^{n}={\mathbf {VDV}}^{-1}{\mathbf {VDV}}^{-1}\ldots {\mathbf {VDV}}^{-1}={\mathbf {VD}}^{n}{\mathbf {V}}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>…</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> <mi mathvariant="bold">D</mi> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> <mi mathvariant="bold">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {A}}^{n}=({\mathbf {VDV}}^{-1})^{n}={\mathbf {VDV}}^{-1}{\mathbf {VDV}}^{-1}\ldots {\mathbf {VDV}}^{-1}={\mathbf {VD}}^{n}{\mathbf {V}}^{-1}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d88bd4917acd685a3648d0769cce69493d5ef4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.384ex; height:3.176ex;" alt="{\displaystyle {\mathbf {A}}^{n}=({\mathbf {VDV}}^{-1})^{n}={\mathbf {VDV}}^{-1}{\mathbf {VDV}}^{-1}\ldots {\mathbf {VDV}}^{-1}={\mathbf {VD}}^{n}{\mathbf {V}}^{-1}}"></noscript><span class="lazy-image-placeholder" style="width: 62.384ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d88bd4917acd685a3648d0769cce69493d5ef4" data-alt="{\displaystyle {\mathbf {A}}^{n}=({\mathbf {VDV}}^{-1})^{n}={\mathbf {VDV}}^{-1}{\mathbf {VDV}}^{-1}\ldots {\mathbf {VDV}}^{-1}={\mathbf {VD}}^{n}{\mathbf {V}}^{-1}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the <a href="/wiki/Matrix_exponential" title="Matrix exponential">matrix exponential</a> eA, a need frequently arising in solving <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equations</a>, <a href="/wiki/Matrix_logarithm" class="mw-redirect" title="Matrix logarithm">matrix logarithms</a> and <a href="/wiki/Square_root_of_a_matrix" title="Square root of a matrix">square roots of matrices</a>.<a href="#cite_note-54">[54]</a> To avoid numerically <a href="/wiki/Condition_number" title="Condition number">ill-conditioned</a> situations, further algorithms such as the <a href="/wiki/Schur_decomposition" title="Schur decomposition">Schur decomposition</a> can be employed.<a href="#cite_note-55">[55]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="Abstract_algebraic_aspects_and_generalizations">Abstract algebraic aspects and generalizations</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=25" title="Edit section: Abstract algebraic aspects and generalizations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> or even <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension is <a href="/wiki/Tensor" title="Tensor">tensors</a>, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realized as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers.<a href="#cite_note-56">[56]</a> Matrices, subject to certain requirements tend to form <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> known as matrix groups. Similarly under certain conditions matrices form <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a> known as <a href="/wiki/Matrix_ring" title="Matrix ring">matrix rings</a>. Though the product of matrices is not in general commutative certain matrices form <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> known as <a href="/wiki/Matrix_field" class="mw-redirect" title="Matrix field">matrix fields</a>. In general, matrices and their <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">multiplication</a> also form a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>, the <a href="/wiki/Category_of_matrices" title="Category of matrices">category of matrices</a>. <div class="mw-heading mw-heading3"><h3 id="Matrices_with_more_general_entries">Matrices with more general entries</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=26" title="Edit section: Matrices with more general entries" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, that is, a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> where <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> operations are defined and well-behaved, may be used instead of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" data-alt="{\displaystyle \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠ or ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" data-alt="{\displaystyle \mathbb {C} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠ for example <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> or <a href="/wiki/Finite_field" title="Finite field">finite fields</a>. For example, <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a> makes use of matrices over finite fields. Wherever <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> are considered, as these are roots of a polynomial they may exist only in a larger field than that of the entries of the matrix; for instance, they may be complex in the case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (for example, to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>, such as ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.325ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff6a3dc2982018ff20f1d2c927afc74a217be6" data-alt="{\displaystyle \mathbb {C} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ⁠ from the outset. More generally, matrices with entries in a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> R are widely used in mathematics.<a href="#cite_note-57">[57]</a> Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(n, R) (also denoted Mn(R)<a href="#cite_note-Pop-Furdui-7">[7]</a>) of all square n-by-n matrices over R is a ring called <a href="/wiki/Matrix_ring" title="Matrix ring">matrix ring</a>, isomorphic to the <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> of the left R-<a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> Rn.<a href="#cite_note-58">[58]</a> If the ring R is <a href="/wiki/Commutative_ring" title="Commutative ring">commutative</a>, that is, its multiplication is commutative, then the ring M(n, R) is also an <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> over R. The <a href="/wiki/Determinant" title="Determinant">determinant</a> of square matrices over a commutative ring R can still be defined using the <a href="/wiki/Leibniz_formula_(determinant)" class="mw-redirect" title="Leibniz formula (determinant)">Leibniz formula</a>; such a matrix is invertible if and only if its determinant is <a href="/wiki/Invertible" class="mw-redirect" title="Invertible">invertible</a> in R, generalizing the situation over a field F, where every nonzero element is invertible.<a href="#cite_note-59">[59]</a> Matrices over <a href="/wiki/Superring" class="mw-redirect" title="Superring">superrings</a> are called <a href="/wiki/Supermatrix" title="Supermatrix">supermatrices</a>.<a href="#cite_note-60">[60]</a> Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is <a href="/wiki/Block_matrix" title="Block matrix">block matrices</a>, which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>; but their sizes must fulfill certain compatibility conditions. <div class="mw-heading mw-heading3"><h3 id="Relationship_to_linear_maps">Relationship to linear maps</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=27" title="Edit section: Relationship to linear maps" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Linear maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cccdefd5f0e00fc2fa5fde2d8cbb039cc408a035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.864ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}"></noscript><span class="lazy-image-placeholder" style="width: 9.864ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cccdefd5f0e00fc2fa5fde2d8cbb039cc408a035" data-alt="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are equivalent to m-by-n matrices, as described <a href="#linear_maps">above</a>. More generally, any linear map f : V → W between finite-<a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">dimensional</a> <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> can be described by a matrix A = (aij), after choosing <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">bases</a> v1, ..., vn of V, and w1, ..., wm of W (so n is the dimension of V and m is the dimension of W), which is such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {v} _{j})=\sum _{i=1}^{m}a_{i,j}\mathbf {w} _{i}\qquad {\mbox{for}}\ j=1,\ldots ,n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="2em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for</mtext> </mstyle> </mrow> <mtext> </mtext> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {v} _{j})=\sum _{i=1}^{m}a_{i,j}\mathbf {w} _{i}\qquad {\mbox{for}}\ j=1,\ldots ,n.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8cc99fbe2677739b89586fcc5ff568c8bae2c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.595ex; height:6.843ex;" alt="{\displaystyle f(\mathbf {v} _{j})=\sum _{i=1}^{m}a_{i,j}\mathbf {w} _{i}\qquad {\mbox{for}}\ j=1,\ldots ,n.}"></noscript><span class="lazy-image-placeholder" style="width: 38.595ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8cc99fbe2677739b89586fcc5ff568c8bae2c8" data-alt="{\displaystyle f(\mathbf {v} _{j})=\sum _{i=1}^{m}a_{i,j}\mathbf {w} _{i}\qquad {\mbox{for}}\ j=1,\ldots ,n.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> In other words, column j of A expresses the image of vj in terms of the basis vectors wI of W; thus this relation uniquely determines the entries of the matrix A. The matrix depends on the choice of the bases: different choices of bases give rise to different, but <a href="/wiki/Matrix_equivalence" title="Matrix equivalence">equivalent matrices</a>.<a href="#cite_note-61">[61]</a> Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the <a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">transpose of the linear map</a> given by A, concerning the <a href="/wiki/Dual_space" title="Dual space">dual bases</a>.<a href="#cite_note-62">[62]</a> These properties can be restated more naturally: the <a href="/wiki/Category_of_matrices" title="Category of matrices">category of matrices</a> with entries in a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  with multiplication as composition is <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalent</a> to the category of finite-dimensional <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> and linear maps over this field.<a href="#cite_note-63">[63]</a> More generally, the set of m×n matrices can be used to represent the R-linear maps between the free modules Rm and Rn for an arbitrary ring R with unity. When n = m composition of these maps is possible, and this gives rise to the <a href="/wiki/Matrix_ring" title="Matrix ring">matrix ring</a> of n×n matrices representing the <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> of Rn. <div class="mw-heading mw-heading3"><h3 id="Matrix_groups">Matrix groups</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=28" title="Edit section: Matrix groups" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> 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/Matrix_group" class="mw-redirect" title="Matrix group">Matrix group</a></div> A <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> is a mathematical structure consisting of a set of objects together with a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a>, that is, an operation combining any two objects to a third, subject to certain requirements.<a href="#cite_note-64">[64]</a> A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group.<a href="#cite_note-65">[65]</a><a href="#cite_note-66">[66]</a> Since a group of every element must be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the <a href="/wiki/General_linear_group" title="General linear group">general linear groups</a>. Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of (that is, a smaller group contained in) their general linear group, called a <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a>.<a href="#cite_note-67">[67]</a> <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">Orthogonal matrices</a>, determined by the condition <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {M}}^{\rm {T}}{\mathbf {M}}={\mathbf {I}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {M}}^{\rm {T}}{\mathbf {M}}={\mathbf {I}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b577e4937fe3414e4fba9fc3830728eb75accb22" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.253ex; height:3.009ex;" alt="{\displaystyle {\mathbf {M}}^{\rm {T}}{\mathbf {M}}={\mathbf {I}},}"></noscript><span class="lazy-image-placeholder" style="width: 11.253ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b577e4937fe3414e4fba9fc3830728eb75accb22" data-alt="{\displaystyle {\mathbf {M}}^{\rm {T}}{\mathbf {M}}={\mathbf {I}},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  form the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>.<a href="#cite_note-68">[68]</a> Every orthogonal matrix has <a href="/wiki/Determinant" title="Determinant">determinant</a> 1 or −1. Orthogonal matrices with determinant 1 form a subgroup called special orthogonal group. Every <a href="/wiki/Finite_group" title="Finite group">finite group</a> is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to a matrix group, as one can see by considering the <a href="/wiki/Regular_representation" title="Regular representation">regular representation</a> of the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a>.<a href="#cite_note-69">[69]</a> General groups can be studied using matrix groups, which are comparatively well understood, using <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>.<a href="#cite_note-70">[70]</a> <div class="mw-heading mw-heading3"><h3 id="Infinite_matrices">Infinite matrices</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=29" title="Edit section: Infinite matrices" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> It is also possible to consider matrices with infinitely many rows and/or columns<a href="#cite_note-71">[71]</a> even though, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers). The basic operations of addition, subtraction, scalar multiplication, and transposition can still be defined without problem; however, matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general. If R is any ring with unity, then the ring of endomorphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\bigoplus _{i\in I}R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\bigoplus _{i\in I}R}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef1aea5d82c1ca687f3345d0108e147c4584f84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.202ex; height:5.676ex;" alt="{\displaystyle M=\bigoplus _{i\in I}R}"></noscript><span class="lazy-image-placeholder" style="width: 11.202ex;height: 5.676ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef1aea5d82c1ca687f3345d0108e147c4584f84" data-alt="{\displaystyle M=\bigoplus _{i\in I}R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as a right R module is isomorphic to the ring of column finite matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {CFM} _{I}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {CFM} _{I}(R)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec68d14ab79cbfeda559749a7b81beeb45010649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.961ex; height:2.843ex;" alt="{\displaystyle \mathrm {CFM} _{I}(R)}"></noscript><span class="lazy-image-placeholder" style="width: 9.961ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec68d14ab79cbfeda559749a7b81beeb45010649" data-alt="{\displaystyle \mathrm {CFM} _{I}(R)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  whose entries are indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\times I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>×</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\times I}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e290ebbc366ef6f8dd83eb6c3d5063f983f1783d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.184ex; height:2.176ex;" alt="{\displaystyle I\times I}"></noscript><span class="lazy-image-placeholder" style="width: 5.184ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e290ebbc366ef6f8dd83eb6c3d5063f983f1783d" data-alt="{\displaystyle I\times I}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and whose columns each contain only finitely many nonzero entries. The endomorphisms of M considered as a left R module result in an analogous object, the row finite matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {RFM} _{I}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {RFM} _{I}(R)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4bf6c26af864a93bbe7f1be107e400074783b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.993ex; height:2.843ex;" alt="{\displaystyle \mathrm {RFM} _{I}(R)}"></noscript><span class="lazy-image-placeholder" style="width: 9.993ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4bf6c26af864a93bbe7f1be107e400074783b8" data-alt="{\displaystyle \mathrm {RFM} _{I}(R)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  whose rows each only have finitely many nonzero entries. If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix A to describe a linear map f : V → W, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector ve of <a href="/wiki/Coefficient" title="Coefficient">coefficients</a>, only finitely many entries vI are nonzero. Now the columns of A describe the images by f of individual basis vectors of V in the basis of W, which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of A however: in the product A · v there are only finitely many nonzero coefficients of v involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover, this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries because each of those columns does. Products of two matrices of the given type are well defined (provided that the column-index and row-index sets match), are of the same type, and correspond to the composition of linear maps. If R is a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">normed</a> ring, then the condition of row or column finiteness can be relaxed. With the norm in place, <a href="/wiki/Absolutely_convergent_series" class="mw-redirect" title="Absolutely convergent series">absolutely convergent series</a> can be used instead of finite sums. For example, the matrices whose column sums are convergent sequences form a ring. Analogously, the matrices whose row sums are convergent series also form a ring. Infinite matrices can also be used to describe <a href="/wiki/Hilbert_space#Operators_on_Hilbert_spaces" title="Hilbert space">operators on Hilbert spaces</a>, where convergence and <a href="/wiki/Continuous_function" title="Continuous function">continuity</a> questions arise, which again results in certain constraints that must be imposed. However, the explicit point of view of matrices tends to obfuscate the matter,<a href="#cite_note-72">[72]</a> and the abstract and more powerful tools of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> can be used instead. <div class="mw-heading mw-heading3"><h3 id="Empty_matrix">Empty matrix</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=30" title="Edit section: Empty matrix" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> An empty matrix is a matrix in which the number of rows or columns (or both) is zero.<a href="#cite_note-73">[73]</a><a href="#cite_note-74">[74]</a> Empty matrices help to deal with maps involving the <a href="/wiki/Zero_vector_space" class="mw-redirect" title="Zero vector space">zero vector space</a>. For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a> allow creating and computing with them. The determinant of the 0-by-0 matrix is 1 as follows regarding the <a href="/wiki/Empty_product" title="Empty product">empty product</a> occurring in the Leibniz formula for the determinant as 1. This value is also consistent with the fact that the identity map from any finite-dimensional space to itself has determinant 1, a fact that is often used as a part of the characterization of determinants. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="Applications">Applications</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=31" title="Edit section: Applications" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in <a href="/wiki/Game_theory" title="Game theory">game theory</a> and <a href="/wiki/Economics" title="Economics">economics</a>, the <a href="/wiki/Payoff_matrix" class="mw-redirect" title="Payoff matrix">payoff matrix</a> encodes the payoff for two players, depending on which out of a given (finite) set of strategies the players choose.<a href="#cite_note-75">[75]</a> <a href="/wiki/Text_mining" title="Text mining">Text mining</a> and automated <a href="/wiki/Thesaurus" title="Thesaurus">thesaurus</a> compilation makes use of <a href="/wiki/Document-term_matrix" title="Document-term matrix">document-term matrices</a> such as <a href="/wiki/Tf-idf" class="mw-redirect" title="Tf-idf">tf-idf</a> to track frequencies of certain words in several documents.<a href="#cite_note-76">[76]</a> Complex numbers can be represented by particular real 2-by-2 matrices via <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+ib\leftrightarrow {\begin{bmatrix}a&-b\\b&a\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> <mo stretchy="false">↔</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mo>−</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+ib\leftrightarrow {\begin{bmatrix}a&-b\\b&a\end{bmatrix}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a953f9863f7860c1a4d31a6db5ab251b96050670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.696ex; height:6.176ex;" alt="{\displaystyle a+ib\leftrightarrow {\begin{bmatrix}a&-b\\b&a\end{bmatrix}},}"></noscript><span class="lazy-image-placeholder" style="width: 19.696ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a953f9863f7860c1a4d31a6db5ab251b96050670" data-alt="{\displaystyle a+ib\leftrightarrow {\begin{bmatrix}a&-b\\b&a\end{bmatrix}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> 1, as <a href="#rotation_matrix">above</a>. A similar interpretation is possible for <a href="/wiki/Quaternion" title="Quaternion">quaternions</a><a href="#cite_note-77">[77]</a> and <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a> in general. Early <a href="/wiki/Encryption" title="Encryption">encryption</a> techniques such as the <a href="/wiki/Hill_cipher" title="Hill cipher">Hill cipher</a> also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.<a href="#cite_note-78">[78]</a> <a href="/wiki/Computer_graphics" title="Computer graphics">Computer graphics</a> uses matrices to represent objects; to calculate transformations of objects using affine <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrices</a> to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation; and to apply image convolutions such as sharpening, blurring, edge detection, and more.<a href="#cite_note-79">[79]</a> Matrices over a <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a> are important in the study of <a href="/wiki/Control_theory" title="Control theory">control theory</a>. <a href="/wiki/Chemistry" title="Chemistry">Chemistry</a> makes use of matrices in various ways, particularly since the use of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum theory</a> to discuss <a href="/wiki/Chemical_bond" title="Chemical bond">molecular bonding</a> and <a href="/wiki/Spectroscopy" title="Spectroscopy">spectroscopy</a>. Examples are the <a href="/wiki/Overlap_matrix" class="mw-redirect" title="Overlap matrix">overlap matrix</a> and the <a href="/wiki/Fock_matrix" title="Fock matrix">Fock matrix</a> used in solving the <a href="/wiki/Roothaan_equations" title="Roothaan equations">Roothaan equations</a> to obtain the <a href="/wiki/Molecular_orbital" title="Molecular orbital">molecular orbitals</a> of the <a href="/wiki/Hartree%E2%80%93Fock_method" title="Hartree–Fock method">Hartree–Fock method</a>. <div class="mw-heading mw-heading3"><h3 id="Graph_theory">Graph theory</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=32" title="Edit section: Graph theory" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Labelled_undirected_graph.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Labelled_undirected_graph.svg/150px-Labelled_undirected_graph.svg.png" decoding="async" width="150" height="137" class="mw-file-element" data-file-width="209" data-file-height="191"></noscript><span class="lazy-image-placeholder" style="width: 150px;height: 137px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Labelled_undirected_graph.svg/150px-Labelled_undirected_graph.svg.png" data-width="150" data-height="137" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Labelled_undirected_graph.svg/225px-Labelled_undirected_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Labelled_undirected_graph.svg/300px-Labelled_undirected_graph.svg.png 2x" data-class="mw-file-element"> </a><figcaption>An undirected graph with adjacency matrix: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&1&0\\1&0&1\\0&1&0\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&1&0\\1&0&1\\0&1&0\end{bmatrix}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51425c53fc279038a1da01e1cece5958d13149b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:12.632ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&1&0\\1&0&1\\0&1&0\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 12.632ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51425c53fc279038a1da01e1cece5958d13149b" data-alt="{\displaystyle {\begin{bmatrix}1&1&0\\1&0&1\\0&1&0\end{bmatrix}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </figcaption></figure> The <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of a <a href="/wiki/Finite_graph" class="mw-redirect" title="Finite graph">finite graph</a> is a basic notion of <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>.<a href="#cite_note-80">[80]</a> It records which vertices of the graph are connected by an edge. Matrices containing just two different values (1 and 0 meaning for example "yes" and "no", respectively) are called <a href="/wiki/Logical_matrix" title="Logical matrix">logical matrices</a>. The <a href="/wiki/Distance_matrix" title="Distance matrix">distance (or cost) matrix</a> contains information about the distances of the edges.<a href="#cite_note-81">[81]</a> These concepts can be applied to <a href="/wiki/Website" title="Website">websites</a> connected by <a href="/wiki/Hyperlink" title="Hyperlink">hyperlinks</a> or cities connected by roads etc., in which case (unless the connection network is extremely dense) the matrices tend to be <a href="/wiki/Sparse_matrix" title="Sparse matrix">sparse</a>, that is, contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in <a href="/wiki/Network_theory" title="Network theory">network theory</a>. <div class="mw-heading mw-heading3"><h3 id="Analysis_and_geometry">Analysis and geometry</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=33" title="Edit section: Analysis and geometry" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a> of a <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306c097f43c91dce633d12cde024948d39e73752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }"></noscript><span class="lazy-image-placeholder" style="width: 11.404ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306c097f43c91dce633d12cde024948d39e73752" data-alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  consists of the <a href="/wiki/Second_derivative" title="Second derivative">second derivatives</a> of ƒ concerning the several coordinate directions, that is,<a href="#cite_note-82">[82]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(f)=\left[{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(f)=\left[{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}\right].}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf91a060a82dd7a47c305e9a4c2865378fcf35f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.967ex; height:6.509ex;" alt="{\displaystyle H(f)=\left[{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}\right].}"></noscript><span class="lazy-image-placeholder" style="width: 19.967ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf91a060a82dd7a47c305e9a4c2865378fcf35f" data-alt="{\displaystyle H(f)=\left[{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}\right].}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Saddle_Point_SVG.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Saddle_Point_SVG.svg/220px-Saddle_Point_SVG.svg.png" decoding="async" width="220" height="165" class="mw-file-element" data-file-width="720" data-file-height="540"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 165px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Saddle_Point_SVG.svg/220px-Saddle_Point_SVG.svg.png" data-width="220" data-height="165" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Saddle_Point_SVG.svg/330px-Saddle_Point_SVG.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Saddle_Point_SVG.svg/440px-Saddle_Point_SVG.svg.png 2x" data-class="mw-file-element"> </a><figcaption>At the <a href="/wiki/Saddle_point" title="Saddle point">saddle point</a> (x = 0, y = 0) (red) of the function f (x,−y) = x2 − y2, the Hessian matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}2&0\\0&-2\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}2&0\\0&-2\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd987cfcc4338b2c3319323e7a73182e09215d09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.662ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}2&0\\0&-2\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 9.662ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd987cfcc4338b2c3319323e7a73182e09215d09" data-alt="{\displaystyle {\begin{bmatrix}2&0\\0&-2\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is <a href="/wiki/Indefinite_matrix" class="mw-redirect" title="Indefinite matrix">indefinite</a>.</figcaption></figure>It encodes information about the local growth behavior of the function: given a <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical point</a> x = (x1, ..., xn), that is, a point where the first <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial f/\partial x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial f/\partial x_{i}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c4abc6fa2414772b9c4e03b6c6e5d0516eaf4f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle \partial f/\partial x_{i}}"></noscript><span class="lazy-image-placeholder" style="width: 7.207ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c4abc6fa2414772b9c4e03b6c6e5d0516eaf4f3" data-alt="{\displaystyle \partial f/\partial x_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  of ƒ vanish, the function has a <a href="/wiki/Local_minimum" class="mw-redirect" title="Local minimum">local minimum</a> if the Hessian matrix is <a href="/wiki/Definiteness_of_a_matrix" class="mw-redirect" title="Definiteness of a matrix">positive definite</a>. <a href="/wiki/Quadratic_programming" title="Quadratic programming">Quadratic programming</a> can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see <a href="#quadratic_forms">above</a>).<a href="#cite_note-83">[83]</a> Another matrix frequently used in geometrical situations is the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobi matrix</a> of a differentiable map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c4ef1546f1443461c62dd06b366da67076e473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.726ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}.}"></noscript><span class="lazy-image-placeholder" style="width: 13.726ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54c4ef1546f1443461c62dd06b366da67076e473" data-alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  If f1, ..., fm denote the components of f, then the Jacobi matrix is defined as<a href="#cite_note-84">[84]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(f)=\left[{\frac {\partial f_{i}}{\partial x_{j}}}\right]_{1\leq i\leq m,1\leq j\leq n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>m</mi> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(f)=\left[{\frac {\partial f_{i}}{\partial x_{j}}}\right]_{1\leq i\leq m,1\leq j\leq n}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbd42114b895c82930ea1e229b566f71fd6b07d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.275ex; height:6.676ex;" alt="{\displaystyle J(f)=\left[{\frac {\partial f_{i}}{\partial x_{j}}}\right]_{1\leq i\leq m,1\leq j\leq n}.}"></noscript><span class="lazy-image-placeholder" style="width: 26.275ex;height: 6.676ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbd42114b895c82930ea1e229b566f71fd6b07d" data-alt="{\displaystyle J(f)=\left[{\frac {\partial f_{i}}{\partial x_{j}}}\right]_{1\leq i\leq m,1\leq j\leq n}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </dd></dl> If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the <a href="/wiki/Implicit_function_theorem" title="Implicit function theorem">implicit function theorem</a>.<a href="#cite_note-85">[85]</a> <a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equations</a> can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic partial differential equations</a> this matrix is positive definite, which has a decisive influence on the set of possible solutions of the equation in question.<a href="#cite_note-86">[86]</a> The <a href="/wiki/Finite_element_method" title="Finite element method">finite element method</a> is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen concerning a sufficiently fine grid, which in turn can be recast as a matrix equation.<a href="#cite_note-87">[87]</a> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Probability_theory_and_statistics">Probability theory and statistics</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=34" title="Edit section: Probability theory and statistics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Markov_chain_SVG.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Markov_chain_SVG.svg/280px-Markov_chain_SVG.svg.png" decoding="async" width="280" height="210" class="mw-file-element" data-file-width="720" data-file-height="540"></noscript><span class="lazy-image-placeholder" style="width: 280px;height: 210px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Markov_chain_SVG.svg/280px-Markov_chain_SVG.svg.png" data-width="280" data-height="210" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Markov_chain_SVG.svg/420px-Markov_chain_SVG.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Markov_chain_SVG.svg/560px-Markov_chain_SVG.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Two different Markov chains. The chart depicts the number of particles (of a total of 1000) in state "2". Both limiting values can be determined from the transition matrices, which are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0.7&0\\0.3&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0.7</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0.7&0\\0.3&1\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d87c47f86479fafe692608300195e0e63631924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.663ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0.7&0\\0.3&1\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 9.663ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d87c47f86479fafe692608300195e0e63631924" data-alt="{\displaystyle {\begin{bmatrix}0.7&0\\0.3&1\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (red) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0.7&0.2\\0.3&0.8\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0.7</mn> </mtd> <mtd> <mn>0.2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.3</mn> </mtd> <mtd> <mn>0.8</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0.7&0.2\\0.3&0.8\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be354953b6cc18f0f2fd70030147f719aade96f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.473ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0.7&0.2\\0.3&0.8\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 11.473ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be354953b6cc18f0f2fd70030147f719aade96f9" data-alt="{\displaystyle {\begin{bmatrix}0.7&0.2\\0.3&0.8\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (black).</figcaption></figure> <a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Stochastic matrices</a> are square matrices whose rows are <a href="/wiki/Probability_vector" title="Probability vector">probability vectors</a>, that is, whose entries are non-negative and sum up to one. Stochastic matrices are used to define <a href="/wiki/Markov_chain" title="Markov chain">Markov chains</a> with finitely many states.<a href="#cite_note-88">[88]</a> A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain-like <a href="/wiki/Absorbing_state" class="mw-redirect" title="Absorbing state">absorbing states</a>, that is, states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.<a href="#cite_note-89">[89]</a> Statistics also makes use of matrices in many different forms.<a href="#cite_note-90">[90]</a> <a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a> is concerned with describing data sets, which can often be represented as <a href="/wiki/Data_matrix_(multivariate_statistics)" class="mw-redirect" title="Data matrix (multivariate statistics)">data matrices</a>, which may then be subjected to <a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">dimensionality reduction</a> techniques. The <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> encodes the mutual <a href="/wiki/Variance" title="Variance">variance</a> of several <a href="/wiki/Random_variable" title="Random variable">random variables</a>.<a href="#cite_note-91">[91]</a> Another technique using matrices are <a href="/wiki/Linear_least_squares" title="Linear least squares">linear least squares</a>, a method that approximates a finite set of pairs (x1, y1), (x2, y2), ..., (xN, yN), by a linear function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}\approx ax_{i}+b,\quad i=1,\ldots ,N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≈</mo> <mi>a</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mspace width="1em"></mspace> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}\approx ax_{i}+b,\quad i=1,\ldots ,N}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4950af44677bf96255e8d2e3fd53b6d5d065a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.896ex; height:2.509ex;" alt="{\displaystyle y_{i}\approx ax_{i}+b,\quad i=1,\ldots ,N}"></noscript><span class="lazy-image-placeholder" style="width: 27.896ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4950af44677bf96255e8d2e3fd53b6d5d065a8" data-alt="{\displaystyle y_{i}\approx ax_{i}+b,\quad i=1,\ldots ,N}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  which can be formulated in terms of matrices, related to the <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> of matrices.<a href="#cite_note-92">[92]</a> <a href="/wiki/Random_matrix" title="Random matrix">Random matrices</a> are matrices whose entries are random numbers, subject to suitable <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a>, such as <a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">matrix normal distribution</a>. Beyond probability theory, they are applied in domains ranging from <a href="/wiki/Number_theory" title="Number theory">number theory</a> to <a href="/wiki/Physics" title="Physics">physics</a>.<a href="#cite_note-93">[93]</a><a href="#cite_note-94">[94]</a> <div class="mw-heading mw-heading3"><h3 id="Symmetries_and_transformations_in_physics">Symmetries and transformations in physics</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=35" title="Edit section: Symmetries and transformations in physics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Symmetry_in_physics" class="mw-redirect" title="Symmetry in physics">Symmetry in physics</a></div> Linear transformations and the associated <a href="/wiki/Symmetry" title="Symmetry">symmetries</a> play a key role in modern physics. For example, <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a> in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> are classified as representations of the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> of special relativity and, more specifically, by their behavior under the <a href="/wiki/Spin_group" title="Spin group">spin group</a>. Concrete representations involving the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> and more general <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a> are an integral part of the physical description of <a href="/wiki/Fermion" title="Fermion">fermions</a>, which behave as <a href="/wiki/Spinor" title="Spinor">spinors</a>.<a href="#cite_note-95">[95]</a> For the three lightest <a href="/wiki/Quark" title="Quark">quarks</a>, there is a group-theoretical representation involving the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a> SU(3); for their calculations, physicists use a convenient matrix representation known as the <a href="/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann matrices</a>, which are also used for the SU(3) <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> that forms the basis of the modern description of strong nuclear interactions, <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>. The <a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa matrix</a>, in turn, expresses the fact that the basic quark states that are important for <a href="/wiki/Weak_interaction" title="Weak interaction">weak interactions</a> are not the same as, but linearly related to the basic quark states that define particles with specific and distinct <a href="/wiki/Mass" title="Mass">masses</a>.<a href="#cite_note-96">[96]</a> <div class="mw-heading mw-heading3"><h3 id="Linear_combinations_of_quantum_states">Linear combinations of quantum states</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=36" title="Edit section: Linear combinations of quantum states" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The first model of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> (<a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a>, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states.<a href="#cite_note-97">[97]</a> This is also referred to as <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a>. One particular example is the <a href="/wiki/Density_matrix" title="Density matrix">density matrix</a> that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" <a href="/wiki/Eigenstates" class="mw-redirect" title="Eigenstates">eigenstates</a>.<a href="#cite_note-98">[98]</a> Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics: Collision reactions such as occur in <a href="/wiki/Particle_accelerator" title="Particle accelerator">particle accelerators</a>, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the <a href="/wiki/S-matrix" title="S-matrix">S-matrix</a>, which encodes all information about the possible interactions between particles.<a href="#cite_note-99">[99]</a> <div class="mw-heading mw-heading3"><h3 id="Normal_modes">Normal modes</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=37" title="Edit section: Normal modes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> A general application of matrices in physics is the description of linearly coupled harmonic systems. The <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a> of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a <a href="/wiki/Force" title="Force">force</a> matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system's <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a>, its <a href="/wiki/Normal_mode" title="Normal mode">normal modes</a>, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of <a href="/wiki/Molecules" class="mw-redirect" title="Molecules">molecules</a>: the internal vibrations of systems consisting of mutually bound component atoms.<a href="#cite_note-100">[100]</a> They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.<a href="#cite_note-101">[101]</a> <div class="mw-heading mw-heading3"><h3 id="Geometrical_optics">Geometrical optics</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=38" title="Edit section: Geometrical optics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <a href="/wiki/Geometrical_optics" title="Geometrical optics">Geometrical optics</a> provides further matrix applications. In this approximative theory, the <a href="/wiki/Light_wave" class="mw-redirect" title="Light wave">wave nature</a> of light is neglected. The result is a model in which <a href="/wiki/Ray_(optics)" title="Ray (optics)">light rays</a> are indeed <a href="/wiki/Ray_(geometry)" class="mw-redirect" title="Ray (geometry)">geometrical rays</a>. If the deflection of light rays by optical elements is small, the action of a <a href="/wiki/Lens_(optics)" class="mw-redirect" title="Lens (optics)">lens</a> or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called <a href="/wiki/Ray_transfer_matrix_analysis" title="Ray transfer matrix analysis">ray transfer matrix analysis</a>: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. There are two kinds of matrices, viz. a refraction matrix describing the refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies. The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components' matrices.<a href="#cite_note-102">[102]</a> <div class="mw-heading mw-heading3"><h3 id="Electronics">Electronics</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=39" title="Edit section: Electronics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Traditional <a href="/wiki/Mesh_analysis" title="Mesh analysis">mesh analysis</a> and <a href="/wiki/Nodal_analysis" title="Nodal analysis">nodal analysis</a> in electronics lead to a system of linear equations that can be described with a matrix. The behavior of many <a href="/wiki/Electronic_component" title="Electronic component">electronic components</a> can be described using matrices. Let A be a 2-dimensional vector with the component's input voltage v1 and input current I1 as its elements, and let B be a 2-dimensional vector with the component's output voltage v2 and output current I2 as its elements. Then the behavior of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a> element (h12), one <a href="/wiki/Admittance" title="Admittance">admittance</a> element (h21), and two <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a> elements (h11 and h22). Calculating a circuit now reduces to multiplying matrices. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"><h2 id="History">History</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=40" title="Edit section: History" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-11 collapsible-block" id="mf-section-11"> Matrices have a long history of application in solving <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a> but they were known as arrays until the 1800s. The <a href="/wiki/Chinese_mathematics" title="Chinese mathematics">Chinese text</a> <a href="/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a> written in the 10th–2nd century BCE is the first example of the use of array methods to solve <a href="/wiki/System_of_linear_equations" title="System of linear equations">simultaneous equations</a>,<a href="#cite_note-103">[103]</a> including the concept of <a href="/wiki/Determinant" title="Determinant">determinants</a>. In 1545 Italian mathematician <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> introduced the method to Europe when he published Ars Magna.<a href="#cite_note-:1-104">[104]</a> The <a href="/wiki/Japanese_mathematics" title="Japanese mathematics">Japanese mathematician</a> <a href="/wiki/Seki_Kowa" class="mw-redirect" title="Seki Kowa">Seki</a> used the same array methods to solve simultaneous equations in 1683.<a href="#cite_note-105">[105]</a> The Dutch mathematician <a href="/wiki/Jan_de_Witt" class="mw-redirect" title="Jan de Witt">Jan de Witt</a> represented transformations using arrays in his 1659 book Elements of Curves (1659).<a href="#cite_note-:0-106">[106]</a> Between 1700 and 1710 <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> publicized the use of arrays for recording information or solutions and experimented with over 50 different systems of arrays.<a href="#cite_note-:1-104">[104]</a> <a href="/wiki/Gabriel_Cramer" title="Gabriel Cramer">Cramer</a> presented <a href="/wiki/Cramer%27s_rule" title="Cramer's rule">his rule</a> in 1750. The term "matrix" (Latin for "womb", "dam" (non-human female animal kept for breeding), "source", "origin", "list", and "register", are derived from <a href="https://en.wiktionary.org/wiki/mater#Latin" class="extiw" title="wikt:mater">mater</a>—mother<a href="#cite_note-107">[107]</a>) was coined by <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">James Joseph Sylvester</a> in 1850,<a href="#cite_note-108">[108]</a> who understood a matrix as an object giving rise to several determinants today called <a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">minors</a>, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 1851 paper, Sylvester explains:<a href="#cite_note-109">[109]</a> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered from the womb of a common parent.</blockquote> <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> published a treatise on geometric transformations using matrices that were not rotated versions of the coefficients being investigated as had previously been done. Instead, he defined operations such as addition, subtraction, multiplication, and division as transformations of those matrices and showed the associative and distributive properties held. Cayley investigated and demonstrated the non-commutative property of matrix multiplication as well as the commutative property of matrix addition.<a href="#cite_note-:1-104">[104]</a> Early matrix theory had limited the use of arrays almost exclusively to determinants and Arthur Cayley's abstract matrix operations were revolutionary. He was instrumental in proposing a matrix concept independent of equation systems. In 1858 <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a> published his A memoir on the theory of matrices<a href="#cite_note-110">[110]</a><a href="#cite_note-111">[111]</a> in which he proposed and demonstrated the <a href="/wiki/Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley–Hamilton theorem</a>.<a href="#cite_note-:1-104">[104]</a> The English mathematician <a href="/wiki/Cuthbert_Edmund_Cullis" title="Cuthbert Edmund Cullis">Cuthbert Edmund Cullis</a> was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation A = [ai,j] to represent a matrix where ai,j refers to the ith row and the jth column.<a href="#cite_note-:1-104">[104]</a> The modern study of determinants sprang from several sources.<a href="#cite_note-112">[112]</a> <a href="/wiki/Number_theory" title="Number theory">Number-theoretical</a> problems led <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a> to relate coefficients of <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a>, that is, expressions such as x2 + xy − 2y2, and <a href="/wiki/Linear_map" title="Linear map">linear maps</a> in three dimensions to matrices. <a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Eisenstein</a> further developed these notions, including the remark that, in modern parlance, <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix products</a> are <a href="/wiki/Non-commutative" class="mw-redirect" title="Non-commutative">non-commutative</a>. <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a> was the first to prove general statements about determinants, using as the definition of the determinant of a matrix A = [ai, j] the following: replace the powers ak j by ajk in the <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}a_{2}\cdots a_{n}\prod _{i<j}(a_{j}-a_{i})\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thickmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}a_{2}\cdots a_{n}\prod _{i<j}(a_{j}-a_{i})\;}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/900f5b2c276b570b623498e7fc64bb845418401c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.334ex; height:5.843ex;" alt="{\displaystyle a_{1}a_{2}\cdots a_{n}\prod _{i<j}(a_{j}-a_{i})\;}"></noscript><span class="lazy-image-placeholder" style="width: 23.334ex;height: 5.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/900f5b2c276b570b623498e7fc64bb845418401c" data-alt="{\displaystyle a_{1}a_{2}\cdots a_{n}\prod _{i<j}(a_{j}-a_{i})\;}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ,</dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∏</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \prod }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423a3226e80f549c55e3873aecbf57af9296e0fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.194ex; height:2.843ex;" alt="{\displaystyle \textstyle \prod }"></noscript><span class="lazy-image-placeholder" style="width: 2.194ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423a3226e80f549c55e3873aecbf57af9296e0fd" data-alt="{\displaystyle \textstyle \prod }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  denotes the <a href="/wiki/Multiplication" title="Multiplication">product</a> of the indicated terms. He also showed, in 1829, that the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of symmetric matrices are real.<a href="#cite_note-113">[113]</a> <a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a> studied "functional determinants"—later called <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobi determinants</a> by Sylvester—which can be used to describe geometric transformations at a local (or <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a>) level, see <a href="#Jacobi_matrix">above</a>. <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Kronecker</a>'s Vorlesungen über die Theorie der Determinanten<a href="#cite_note-114">[114]</a> and <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass'</a> Zur Determinantentheorie,<a href="#cite_note-115">[115]</a> both published in 1903, first treated determinants <a href="/wiki/Axiom" title="Axiom">axiomatically</a>, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established. Many theorems were first established for small matrices only, for example, the <a href="/wiki/Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley–Hamilton theorem</a> was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a> for 4×4 matrices. <a href="/wiki/Georg_Frobenius" class="mw-redirect" title="Georg Frobenius">Frobenius</a>, working on <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear forms</a>, generalized the theorem to all dimensions (1898). Also at the end of the 19th century, the <a href="/wiki/Gauss%E2%80%93Jordan_elimination" class="mw-redirect" title="Gauss–Jordan elimination">Gauss–Jordan elimination</a> (generalizing a special case now known as <a href="/wiki/Gauss_elimination" class="mw-redirect" title="Gauss elimination">Gauss elimination</a>) was established by <a href="/wiki/Wilhelm_Jordan_(geodesist)" title="Wilhelm Jordan (geodesist)">Wilhelm Jordan</a>. In the early 20th century, matrices attained a central role in linear algebra,<a href="#cite_note-116">[116]</a> partially due to their use in the classification of the <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex number</a> systems of the previous century. The inception of <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> by <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a>, <a href="/wiki/Max_Born" title="Max Born">Born</a> and <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a> led to studying matrices with infinitely many rows and columns.<a href="#cite_note-117">[117]</a> Later, <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a> carried out the <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">mathematical formulation of quantum mechanics</a>, by further developing <a href="/wiki/Functional_analysis" title="Functional analysis">functional analytic</a> notions such as <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a> on <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>, which, very roughly speaking, correspond to <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, but with an infinity of <a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">independent directions</a>. <div class="mw-heading mw-heading3"><h3 id='Other_historical_usages_of_the_word_"matrix"_in_mathematics'>Other historical usages of the word "matrix" in mathematics</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=41" title='Edit section: Other historical usages of the word "matrix" in mathematics' class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The word has been used in unusual ways by at least two authors of historical importance. <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> and <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> in their <a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a> (1910–1913) use the word "matrix" in the context of their <a href="/wiki/Axiom_of_reducibility" title="Axiom of reducibility">axiom of reducibility</a>. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the "bottom" (0 order) the function is identical to its <a href="/wiki/Extension_(predicate_logic)" title="Extension (predicate logic)">extension</a>:<a href="#cite_note-118">[118]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">Let us give the name of matrix to any function, of however many variables, that does not involve any <a href="/wiki/Apparent_variable" class="mw-redirect" title="Apparent variable">apparent variables</a>. Then, any possible function other than a matrix derives from a matrix using generalization, that is, by considering the proposition that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined.</blockquote> For example, a function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, for example, y, by "considering" the function for all possible values of "individuals" ai substituted in place of a variable x. And then the resulting collection of functions of the single variable y, that is, ∀ai: Φ(ai, y), can be reduced to a "matrix" of values by "considering" the function for all possible values of "individuals" bi substituted in place of variable y: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall b_{j}\forall a_{i}:\phi (a_{i},b_{j}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi mathvariant="normal">∀</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall b_{j}\forall a_{i}:\phi (a_{i},b_{j}).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f21a192144aae927542bc2a3804dba168d3d89dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.271ex; height:3.009ex;" alt="{\displaystyle \forall b_{j}\forall a_{i}:\phi (a_{i},b_{j}).}"></noscript><span class="lazy-image-placeholder" style="width: 17.271ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f21a192144aae927542bc2a3804dba168d3d89dd" data-alt="{\displaystyle \forall b_{j}\forall a_{i}:\phi (a_{i},b_{j}).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> in his 1946 Introduction to Logic used the word "matrix" synonymously with the notion of <a href="/wiki/Truth_table" title="Truth table">truth table</a> as used in mathematical logic.<a href="#cite_note-119">[119]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(12)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=42" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-12 collapsible-block" id="mf-section-12"> <style data-mw-deduplicate="TemplateStyles:r1259569809">.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"><noscript><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" data-file-width="128" data-file-height="128"></noscript><span class="lazy-image-placeholder" style="width: 28px;height: 28px;" data-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" data-alt="icon" data-width="28" data-height="28" data-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-class="mw-file-element"> </a><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> <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" style="column-width: 30em;"> <ul><li><a href="/wiki/List_of_named_matrices" title="List of named matrices">List of named matrices</a></li> <li><a href="/wiki/Algebraic_multiplicity" class="mw-redirect" title="Algebraic multiplicity">Algebraic multiplicity</a> – Multiplicity of an eigenvalue as a root of the characteristic polynomial</li> <li><a href="/wiki/Geometric_multiplicity" class="mw-redirect" title="Geometric multiplicity">Geometric multiplicity</a> – Dimension of the eigenspace associated with an eigenvalue</li> <li><a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a> – Orthonormalization of a set of vectors</li> <li><a href="/wiki/Irregular_matrix" title="Irregular matrix">Irregular matrix</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix calculus</a> – Specialized notation for multivariable calculus</li> <li><a href="/wiki/Matrix_function" class="mw-redirect" title="Matrix function">Matrix function</a> – Function that maps matrices to matricesPages displaying short descriptions of redirect targets</li> <li><a href="/wiki/Matrix_multiplication_algorithm" title="Matrix multiplication algorithm">Matrix multiplication algorithm</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a> — A generalization of matrices with any number of indices</li> <li><a href="/wiki/Bohemian_matrices" title="Bohemian matrices">Bohemian matrices</a> – Set of matrices</li> <li><a href="/wiki/Category_of_matrices" title="Category of matrices">Category of matrices</a> — The algebraic structure formed by matrices and their multiplication</li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(13)"><h2 id="Notes">Notes</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=43" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-13 collapsible-block" id="mf-section-13"> <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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> However, in the case of adjacency matrices, <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> or a variant of it allows the simultaneous computation of the number of paths between any two vertices, and of the shortest length of a path between two vertices. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> Lang <a href="#CITEREFLang2002">2002</a> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFFraleigh1976">Fraleigh (1976</a>, p. 209) </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFNering1970">Nering (1970</a>, p. 37) </li> <li id="cite_note-:4-5">^ <a href="#cite_ref-:4_5-0">a</a> <a href="#cite_ref-:4_5-1">b</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="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Matrix.html">"Matrix"</a>. mathworld.wolfram.com. Retrieved 2020-08-19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Matrix&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMatrix.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> Oualline <a href="#CITEREFOualline2003">2003</a>, Ch. 5 </li> <li id="cite_note-Pop-Furdui-7">^ <a href="#cite_ref-Pop-Furdui_7-0">a</a> <a href="#cite_ref-Pop-Furdui_7-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPopFurdui2017" class="citation book cs1">Pop; Furdui (2017). Square Matrices of Order 2. Springer International Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-54938-5" title="Special:BookSources/978-3-319-54938-5"><bdi>978-3-319-54938-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Square+Matrices+of+Order+2&rft.pub=Springer+International+Publishing&rft.date=2017&rft.isbn=978-3-319-54938-5&rft.au=Pop&rft.au=Furdui&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose) </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Theorem I.2.6 </li> <li id="cite_note-:5-10">^ <a href="#cite_ref-:5_10-0">a</a> <a href="#cite_ref-:5_10-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathsisfun.com/algebra/matrix-multiplying.html">"How to Multiply Matrices"</a>. www.mathsisfun.com. Retrieved 2020-08-19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft.atitle=How+to+Multiply+Matrices&rft_id=https%3A%2F%2Fmathsisfun.com%2Falgebra%2Fmatrix-multiplying.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Definition I.2.20 </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Theorem I.2.24 </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> Horn & Johnson <a href="#CITEREFHornJohnson1985">1985</a>, Ch. 4 and 5 </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFBronson1970">Bronson (1970</a>, p. 16) </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFKreyszig1972">Kreyszig (1972</a>, p. 220) </li> <li id="cite_note-Protter_1970_869-16">^ <a href="#cite_ref-Protter_1970_869_16-0">a</a> <a href="#cite_ref-Protter_1970_869_16-1">b</a> <a href="#CITEREFProtterMorrey1970">Protter & Morrey (1970</a>, p. 869) </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFKreyszig1972">Kreyszig (1972</a>, pp. 241, 244) </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchneiderBarker2012" class="citation cs2">Schneider, Hans; Barker, George Phillip (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9vjBAgAAQBAJ&pg=PA251">Matrices and Linear Algebra</a>, Dover Books on Mathematics, Courier Dover Corporation, p. 251, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-13930-2" title="Special:BookSources/978-0-486-13930-2"><bdi>978-0-486-13930-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrices+and+Linear+Algebra&rft.series=Dover+Books+on+Mathematics&rft.pages=251&rft.pub=Courier+Dover+Corporation&rft.date=2012&rft.isbn=978-0-486-13930-2&rft.aulast=Schneider&rft.aufirst=Hans&rft.au=Barker%2C+George+Phillip&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9vjBAgAAQBAJ%26pg%3DPA251&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988">. </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerlis1991" class="citation cs2">Perlis, Sam (1991), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5_sxtcnvLhoC&pg=PA103">Theory of Matrices</a>, Dover books on advanced mathematics, Courier Dover Corporation, p. 103, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-66810-9" title="Special:BookSources/978-0-486-66810-9"><bdi>978-0-486-66810-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Matrices&rft.series=Dover+books+on+advanced+mathematics&rft.pages=103&rft.pub=Courier+Dover+Corporation&rft.date=1991&rft.isbn=978-0-486-66810-9&rft.aulast=Perlis&rft.aufirst=Sam&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5_sxtcnvLhoC%26pg%3DPA103&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988">. </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnton2010" class="citation cs2">Anton, Howard (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YmcQJoFyZ5gC&pg=PA414">Elementary Linear Algebra</a> (10th ed.), John Wiley & Sons, p. 414, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-45821-1" title="Special:BookSources/978-0-470-45821-1"><bdi>978-0-470-45821-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra&rft.pages=414&rft.edition=10th&rft.pub=John+Wiley+%26+Sons&rft.date=2010&rft.isbn=978-0-470-45821-1&rft.aulast=Anton&rft.aufirst=Howard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DYmcQJoFyZ5gC%26pg%3DPA414&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988">. </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornJohnson2012" class="citation cs2">Horn, Roger A.; Johnson, Charles R. (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5I5AYeeh0JUC&pg=PA17">Matrix Analysis</a> (2nd ed.), Cambridge University Press, p. 17, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83940-2" title="Special:BookSources/978-0-521-83940-2"><bdi>978-0-521-83940-2</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+Analysis&rft.pages=17&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2012&rft.isbn=978-0-521-83940-2&rft.aulast=Horn&rft.aufirst=Roger+A.&rft.au=Johnson%2C+Charles+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5I5AYeeh0JUC%26pg%3DPA17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988">. </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, I.2.21 and 22 </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> Greub <a href="#CITEREFGreub1975">1975</a>, Section III.2 </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Definition II.3.3 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> Greub <a href="#CITEREFGreub1975">1975</a>, Section III.1 </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Theorem II.3.22 </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> Horn & Johnson <a href="#CITEREFHornJohnson1985">1985</a>, Theorem 2.5.6 </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Definition I.2.28 </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Definition I.5.13 </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> Horn & Johnson <a href="#CITEREFHornJohnson1985">1985</a>, Chapter 7 </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> Horn & Johnson <a href="#CITEREFHornJohnson1985">1985</a>, Theorem 7.2.1 </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> Horn & Johnson <a href="#CITEREFHornJohnson1985">1985</a>, Example 4.0.6, p. 169 </li> <li id="cite_note-:3-33"><a href="#cite_ref-:3_33-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://britannica.com/science/matrix-mathematics">"Matrix | mathematics"</a>. Encyclopedia Britannica. Retrieved 2020-08-19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+Britannica&rft.atitle=Matrix+%7C+mathematics&rft_id=https%3A%2F%2Fbritannica.com%2Fscience%2Fmatrix-mathematics&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Definition III.2.1 </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Theorem III.2.12 </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Corollary III.2.16 </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> Mirsky <a href="#CITEREFMirsky1990">1990</a>, Theorem 1.4.1 </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Theorem III.3.18 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> Eigen means "own" in <a href="/wiki/German_language" title="German language">German</a> and in <a href="/wiki/Dutch_language" title="Dutch language">Dutch</a>. </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Definition III.4.1 </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Definition III.4.9 </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> Brown <a href="#CITEREFBrown1991">1991</a>, Corollary III.4.10 </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> Householder <a href="#CITEREFHouseholder1975">1975</a>, Ch. 7 </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> Bau III & Trefethen <a href="#CITEREFBau_IIITrefethen1997">1997</a> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> Golub & Van Loan <a href="#CITEREFGolubVan_Loan1996">1996</a>, Algorithm 1.3.1 </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> Golub & Van Loan <a href="#CITEREFGolubVan_Loan1996">1996</a>, Chapters 9 and 10, esp. section 10.2 </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> Golub & Van Loan <a href="#CITEREFGolubVan_Loan1996">1996</a>, Chapter 2.3 </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrcar2011" class="citation journal cs1">Grcar, Joseph F. (2011-01-01). <a rel="nofollow" class="external text" href="https://epubs.siam.org/doi/10.1137/080734716">"John von Neumann's Analysis of Gaussian Elimination and the Origins of Modern Numerical Analysis"</a>. SIAM Review. 53 (4): 607–682. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F080734716">10.1137/080734716</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/0036-1445">0036-1445</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Review&rft.atitle=John+von+Neumann%27s+Analysis+of+Gaussian+Elimination+and+the+Origins+of+Modern+Numerical+Analysis&rft.volume=53&rft.issue=4&rft.pages=607-682&rft.date=2011-01-01&rft_id=info%3Adoi%2F10.1137%2F080734716&rft.issn=0036-1445&rft.aulast=Grcar&rft.aufirst=Joseph+F.&rft_id=https%3A%2F%2Fepubs.siam.org%2Fdoi%2F10.1137%2F080734716&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> For example, <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>, see Wolfram <a href="#CITEREFWolfram2003">2003</a>, Ch. 3.7 </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> Press, Flannery & Teukolsky et al. <a href="#CITEREFPressFlanneryTeukolskyVetterling1992">1992</a> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> Stoer & Bulirsch <a href="#CITEREFStoerBulirsch2002">2002</a>, Section 4.1 </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> Horn & Johnson <a href="#CITEREFHornJohnson1985">1985</a>, Theorem 2.5.4 </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> Horn & Johnson <a href="#CITEREFHornJohnson1985">1985</a>, Ch. 3.1, 3.2 </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> Arnold & Cooke <a href="#CITEREFArnoldCooke1992">1992</a>, Sections 14.5, 7, 8 </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> Bronson <a href="#CITEREFBronson1989">1989</a>, Ch. 15 </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> Coburn <a href="#CITEREFCoburn1955">1955</a>, Ch. V </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> Lang <a href="#CITEREFLang2002">2002</a>, Chapter XIII </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> Lang <a href="#CITEREFLang2002">2002</a>, XVII.1, p. 643 </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> Lang <a href="#CITEREFLang2002">2002</a>, Proposition XIII.4.16 </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> Reichl <a href="#CITEREFReichl2004">2004</a>, Section L.2 </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> Greub <a href="#CITEREFGreub1975">1975</a>, Section III.3 </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> Greub <a href="#CITEREFGreub1975">1975</a>, Section III.3.13 </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <a href="#CITEREFPerrone2024">Perrone (2024)</a>, pp. 99–100 </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> See any standard reference in a group. </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> Additionally, the group must be <a href="/wiki/Closed_subset" class="mw-redirect" title="Closed subset">closed</a> in the general linear group. </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> Baker <a href="#CITEREFBaker2003">2003</a>, Def. 1.30 </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> Baker <a href="#CITEREFBaker2003">2003</a>, Theorem 1.2 </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> Artin <a href="#CITEREFArtin1991">1991</a>, Chapter 4.5 </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> Rowen <a href="#CITEREFRowen2008">2008</a>, Example 19.2, p. 198 </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> See any reference in representation theory or <a href="/wiki/Group_representation" title="Group representation">group representation</a>. </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> See the item "Matrix" in Itõ, ed. <a href="#CITEREFIt%C3%B51987">1987</a> </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> "Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps." Halmos <a href="#CITEREFHalmos1982">1982</a>, p. 23, Chapter 5 </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> "Empty Matrix: A matrix is empty if either its row or column dimension is zero", <a rel="nofollow" class="external text" href="https://omatrix.com/manual/glossary.htm">Glossary</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090429015728/http://www.omatrix.com/manual/glossary.htm">Archived</a> 2009-04-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, O-Matrix v6 User Guide </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> "A matrix having at least one dimension equal to zero is called an empty matrix", <a rel="nofollow" class="external text" href="https://system.nada.kth.se/unix/software/matlab/Release_14.1/techdoc/matlab_prog/ch_dat29.html">MATLAB Data Structures</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091228102653/http://www.system.nada.kth.se/unix/software/matlab/Release_14.1/techdoc/matlab_prog/ch_dat29.html">Archived</a> 2009-12-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> Fudenberg & Tirole <a href="#CITEREFFudenbergTirole1983">1983</a>, Section 1.1.1 </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> Manning <a href="#CITEREFManning1999">1999</a>, Section 15.3.4 </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> Ward <a href="#CITEREFWard1997">1997</a>, Ch. 2.8 </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> Stinson <a href="#CITEREFStinson2005">2005</a>, Ch. 1.1.5 and 1.2.4 </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> Association for Computing Machinery <a href="#CITEREFAssociation_for_Computing_Machinery1979">1979</a>, Ch. 7 </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> Godsil & Royle <a href="#CITEREFGodsilRoyle2004">2004</a>, Ch. 8.1 </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> Punnen <a href="#CITEREFPunnen2002">2002</a> </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> Lang <a href="#CITEREFLang1987a">1987a</a>, Ch. XVI.6 </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> Nocedal <a href="#CITEREFNocedal2006">2006</a>, Ch. 16 </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> Lang <a href="#CITEREFLang1987a">1987a</a>, Ch. XVI.1 </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> Lang <a href="#CITEREFLang1987a">1987a</a>, Ch. XVI.5. For a more advanced, and more general statement see Lang <a href="#CITEREFLang1969">1969</a>, Ch. VI.2 </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> Gilbarg & Trudinger <a href="#CITEREFGilbargTrudinger2001">2001</a> </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> Šolin <a href="#CITEREF%C5%A0olin2005">2005</a>, Ch. 2.5. See also <a href="/wiki/Stiffness_method" class="mw-redirect" title="Stiffness method">stiffness method</a>. </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> Latouche & Ramaswami <a href="#CITEREFLatoucheRamaswami1999">1999</a> </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> Mehata & Srinivasan <a href="#CITEREFMehataSrinivasan1978">1978</a>, Ch. 2.8 </li> <li id="cite_note-90"><a href="#cite_ref-90">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHealy1986" class="citation cs2"><a href="/wiki/Michael_Healy_(statistician)" title="Michael Healy (statistician)">Healy, Michael</a> (1986), Matrices for Statistics, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850702-4" title="Special:BookSources/978-0-19-850702-4"><bdi>978-0-19-850702-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrices+for+Statistics&rft.pub=Oxford+University+Press&rft.date=1986&rft.isbn=978-0-19-850702-4&rft.aulast=Healy&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> Krzanowski <a href="#CITEREFKrzanowski1988">1988</a>, Ch. 2.2., p. 60 </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> Krzanowski <a href="#CITEREFKrzanowski1988">1988</a>, Ch. 4.1 </li> <li id="cite_note-93"><a href="#cite_ref-93">^</a> <a href="/wiki/Brian_Conrey" title="Brian Conrey">Conrey</a> <a href="#CITEREFConrey2007">2007</a> </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> Zabrodin, Brezin & Kazakov et al. <a href="#CITEREFZabrodinBrezinKazakovSerban2006">2006</a> </li> <li id="cite_note-95"><a href="#cite_ref-95">^</a> Itzykson & Zuber <a href="#CITEREFItzyksonZuber1980">1980</a>, Ch. 2 </li> <li id="cite_note-96"><a href="#cite_ref-96">^</a> see Burgess & Moore <a href="#CITEREFBurgessMoore2007">2007</a>, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi–Maskawa matrix) </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> Schiff <a href="#CITEREFSchiff1968">1968</a>, Ch. 6 </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> Bohm <a href="#CITEREFBohm2001">2001</a>, sections II.4 and II.8 </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> Weinberg <a href="#CITEREFWeinberg1995">1995</a>, Ch. 3 </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> Wherrett <a href="#CITEREFWherrett1987">1987</a>, part II </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> Riley, Hobson & Bence <a href="#CITEREFRileyHobsonBence1997">1997</a>, 7.17 </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> Guenther <a href="#CITEREFGuenther1990">1990</a>, Ch. 5 </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> Shen, Crossley & Lun <a href="#CITEREFShenCrossleyLun1999">1999</a> cited by Bretscher <a href="#CITEREFBretscher2005">2005</a>, p. 1 </li> <li id="cite_note-:1-104">^ <a href="#cite_ref-:1_104-0">a</a> <a href="#cite_ref-:1_104-1">b</a> <a href="#cite_ref-:1_104-2">c</a> <a href="#cite_ref-:1_104-3">d</a> <a href="#cite_ref-:1_104-4">e</a> Discrete Mathematics 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 <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/978-0-321-07912-1" title="Special:BookSources/978-0-321-07912-1">978-0-321-07912-1</a>, p. 564-565 </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeedhamWang_Ling1959" class="citation book cs1"><a href="/wiki/Joseph_Needham" title="Joseph Needham">Needham, Joseph</a>; <a href="/wiki/Wang_Ling_(historian)" title="Wang Ling (historian)">Wang Ling</a> (1959). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jfQ9E0u4pLAC&pg=PA117">Science and Civilisation in China</a>. Vol. III. Cambridge: Cambridge University Press. p. 117. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-05801-8" title="Special:BookSources/978-0-521-05801-8"><bdi>978-0-521-05801-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+and+Civilisation+in+China&rft.place=Cambridge&rft.pages=117&rft.pub=Cambridge+University+Press&rft.date=1959&rft.isbn=978-0-521-05801-8&rft.aulast=Needham&rft.aufirst=Joseph&rft.au=Wang+Ling&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjfQ9E0u4pLAC%26pg%3DPA117&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-:0-106"><a href="#cite_ref-:0_106-0">^</a> Discrete Mathematics 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 <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/978-0-321-07912-1" title="Special:BookSources/978-0-321-07912-1">978-0-321-07912-1</a>, p. 564 </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://merriam-webster.com/dictionary/matrix">Merriam-Webster dictionary</a>, Merriam-Webster, retrieved April 20, 2009</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Merriam-Webster+dictionary&rft.pub=Merriam-Webster&rft_id=https%3A%2F%2Fmerriam-webster.com%2Fdictionary%2Fmatrix&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> Although many sources state that J. J. Sylvester coined the mathematical term "matrix" in 1848, Sylvester published nothing in 1848. (For proof that Sylvester published nothing in 1848, see J. J. Sylvester with H. F. Baker, ed., The Collected Mathematical Papers of James Joseph Sylvester (Cambridge, England: Cambridge University Press, 1904), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=r-kZAQAAIAAJ&pg=PR6">vol. 1.</a>) His earliest use of the term "matrix" occurs in 1850 in J. J. Sylvester (1850) "Additions to the articles in the September number of this journal, "On a new class of theorems," and on Pascal's theorem," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 37: 363-370. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CBhDAQAAIAAJ&pg=PA369">From page 369</a>: "For this purpose, we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This does not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants ... " </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5GQPlxWrDiEC&pg=PA247">Paper 37</a>, p. 247 </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> Phil.Trans. 1858, vol.148, pp.17-37 Math. Papers II 475-496 </li> <li id="cite_note-111"><a href="#cite_ref-111">^</a> Dieudonné, ed. <a href="#CITEREFDieudonn%C3%A91978">1978</a>, Vol. 1, Ch. III, p. 96 </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> Knobloch <a href="#CITEREFKnobloch1994">1994</a> </li> <li id="cite_note-113"><a href="#cite_ref-113">^</a> Hawkins <a href="#CITEREFHawkins1975">1975</a> </li> <li id="cite_note-114"><a href="#cite_ref-114">^</a> Kronecker <a href="#CITEREFKronecker1897">1897</a> </li> <li id="cite_note-115"><a href="#cite_ref-115">^</a> Weierstrass <a href="#CITEREFWeierstrass1915">1915</a>, pp. 271–286 </li> <li id="cite_note-116"><a href="#cite_ref-116">^</a> Bôcher <a href="#CITEREFB%C3%B4cher2004">2004</a> </li> <li id="cite_note-117"><a href="#cite_ref-117">^</a> Mehra & Rechenberg <a href="#CITEREFMehraRechenberg1987">1987</a> </li> <li id="cite_note-118"><a href="#cite_ref-118">^</a> Whitehead, Alfred North; and Russell, Bertrand (1913) Principia Mathematica to *56, Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff. </li> <li id="cite_note-119"><a href="#cite_ref-119">^</a> Tarski, Alfred; (1946) Introduction to Logic and the Methodology of Deductive Sciences, Dover Publications, Inc, New York NY, <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/0-486-28462-X" title="Special:BookSources/0-486-28462-X">0-486-28462-X</a>. </li> </ol></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width reflist-columns-3"> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(14)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=44" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-14 collapsible-block" id="mf-section-14"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnton1987" class="citation cs2">Anton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-84819-0" title="Special:BookSources/0-471-84819-0"><bdi>0-471-84819-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra&rft.place=New+York&rft.edition=5th&rft.pub=Wiley&rft.date=1987&rft.isbn=0-471-84819-0&rft.aulast=Anton&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArnoldCooke1992" class="citation cs2"><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Arnold, Vladimir I.</a>; <a href="/w/index.php?title=Roger_Cooke_(mathematician)&action=edit&redlink=1" class="new" title="Roger Cooke (mathematician) (page does not exist)">Cooke, Roger</a> (1992), Ordinary differential equations, Berlin, DE; New York, NY: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-54813-3" title="Special:BookSources/978-3-540-54813-3"><bdi>978-3-540-54813-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ordinary+differential+equations&rft.place=Berlin%2C+DE%3B+New+York%2C+NY&rft.pub=Springer-Verlag&rft.date=1992&rft.isbn=978-3-540-54813-3&rft.aulast=Arnold&rft.aufirst=Vladimir+I.&rft.au=Cooke%2C+Roger&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArtin1991" class="citation cs2"><a href="/wiki/Michael_Artin" title="Michael Artin">Artin, Michael</a> (1991), Algebra, <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-510-1" title="Special:BookSources/978-0-89871-510-1"><bdi>978-0-89871-510-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pub=Prentice+Hall&rft.date=1991&rft.isbn=978-0-89871-510-1&rft.aulast=Artin&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAssociation_for_Computing_Machinery1979" class="citation cs2">Association for Computing Machinery (1979), Computer Graphics, Tata McGraw–Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-059376-3" title="Special:BookSources/978-0-07-059376-3"><bdi>978-0-07-059376-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Graphics&rft.pub=Tata+McGraw%E2%80%93Hill&rft.date=1979&rft.isbn=978-0-07-059376-3&rft.au=Association+for+Computing+Machinery&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaker2003" class="citation cs2">Baker, Andrew J. (2003), <a rel="nofollow" class="external text" href="https://archive.org/details/matrixgroupsintr0000bake">Matrix Groups: An Introduction to Lie Group Theory</a>, Berlin, DE; New York, NY: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-470-3" title="Special:BookSources/978-1-85233-470-3"><bdi>978-1-85233-470-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Groups%3A+An+Introduction+to+Lie+Group+Theory&rft.place=Berlin%2C+DE%3B+New+York%2C+NY&rft.pub=Springer-Verlag&rft.date=2003&rft.isbn=978-1-85233-470-3&rft.aulast=Baker&rft.aufirst=Andrew+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmatrixgroupsintr0000bake&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBau_IIITrefethen1997" class="citation cs2">Bau III, David; <a href="/wiki/Lloyd_N._Trefethen" class="mw-redirect" title="Lloyd N. Trefethen">Trefethen, Lloyd N.</a> (1997), Numerical linear algebra, Philadelphia, PA: Society for Industrial and Applied Mathematics, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-361-9" title="Special:BookSources/978-0-89871-361-9"><bdi>978-0-89871-361-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+linear+algebra&rft.place=Philadelphia%2C+PA&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1997&rft.isbn=978-0-89871-361-9&rft.aulast=Bau+III&rft.aufirst=David&rft.au=Trefethen%2C+Lloyd+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeauregardFraleigh1973" class="citation cs2">Beauregard, Raymond A.; Fraleigh, John B. 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Golub">Golub, Gene H.</a>; <a href="/wiki/Charles_F._Van_Loan" title="Charles F. Van Loan">Van Loan, Charles F.</a> (1996), Matrix Computations (3rd ed.), Johns Hopkins, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-5414-9" title="Special:BookSources/978-0-8018-5414-9"><bdi>978-0-8018-5414-9</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+Computations&rft.edition=3rd&rft.pub=Johns+Hopkins&rft.date=1996&rft.isbn=978-0-8018-5414-9&rft.aulast=Golub&rft.aufirst=Gene+H.&rft.au=Van+Loan%2C+Charles+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreub1975" class="citation cs2">Greub, Werner Hildbert (1975), Linear algebra, Graduate Texts in Mathematics, Berlin, DE; New York, NY: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90110-7" title="Special:BookSources/978-0-387-90110-7"><bdi>978-0-387-90110-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+algebra&rft.place=Berlin%2C+DE%3B+New+York%2C+NY&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1975&rft.isbn=978-0-387-90110-7&rft.aulast=Greub&rft.aufirst=Werner+Hildbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1982" class="citation cs2"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul Richard</a> (1982), A Hilbert space problem book, Graduate Texts in Mathematics, vol. 19 (2nd ed.), Berlin, DE; New York, NY: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90685-0" title="Special:BookSources/978-0-387-90685-0"><bdi>978-0-387-90685-0</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=0675952">0675952</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Hilbert+space+problem+book&rft.place=Berlin%2C+DE%3B+New+York%2C+NY&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1982&rft.isbn=978-0-387-90685-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D675952%23id-name%3DMR&rft.aulast=Halmos&rft.aufirst=Paul+Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornJohnson1985" class="citation cs2"><a href="/wiki/Roger_Horn" title="Roger Horn">Horn, Roger A.</a>; <a href="/wiki/Charles_Royal_Johnson" title="Charles Royal Johnson">Johnson, Charles R.</a> (1985), Matrix Analysis, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-38632-6" title="Special:BookSources/978-0-521-38632-6"><bdi>978-0-521-38632-6</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+Analysis&rft.pub=Cambridge+University+Press&rft.date=1985&rft.isbn=978-0-521-38632-6&rft.aulast=Horn&rft.aufirst=Roger+A.&rft.au=Johnson%2C+Charles+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHouseholder1975" class="citation cs2">Householder, Alston S. 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(1992), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090906113144/http://www.mpi-hd.mpg.de/astrophysik/HEA/internal/Numerical_Recipes/f2-3.pdf">"LU Decomposition and Its Applications"</a> (PDF), Numerical Recipes in FORTRAN: The Art of Scientific Computing (2nd ed.), Cambridge University Press, pp. 34–42, archived from the original on 2009-09-06</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=LU+Decomposition+and+Its+Applications&rft.btitle=Numerical+Recipes+in+FORTRAN%3A+The+Art+of+Scientific+Computing&rft.pages=34-42&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1992&rft.aulast=Press&rft.aufirst=William+H.&rft.au=Flannery%2C+Brian+P.&rft.au=Teukolsky%2C+Saul+A.&rft.au=Vetterling%2C+William+T.&rft_id=https%3A%2F%2Fmpi-hd.mpg.de%2Fastrophysik%2FHEA%2Finternal%2FNumerical_Recipes%2Ff2-3.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: unfit URL (<a href="/wiki/Category:CS1_maint:_unfit_URL" title="Category:CS1 maint: unfit URL">link</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProtterMorrey1970" class="citation cs2">Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/76087042">76087042</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+Calculus+with+Analytic+Geometry&rft.place=Reading&rft.edition=2nd&rft.pub=Addison-Wesley&rft.date=1970&rft_id=info%3Alccn%2F76087042&rft.aulast=Protter&rft.aufirst=Murray+H.&rft.au=Morrey%2C+Charles+B.+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPunnenGutin2002" class="citation cs2">Punnen, Abraham P.; Gutin, Gregory (2002), The traveling salesman problem and its variations, Boston, MA: Kluwer Academic Publishers, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-0664-7" title="Special:BookSources/978-1-4020-0664-7"><bdi>978-1-4020-0664-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+traveling+salesman+problem+and+its+variations&rft.place=Boston%2C+MA&rft.pub=Kluwer+Academic+Publishers&rft.date=2002&rft.isbn=978-1-4020-0664-7&rft.aulast=Punnen&rft.aufirst=Abraham+P.&rft.au=Gutin%2C+Gregory&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReichl2004" class="citation cs2"><a href="/wiki/Linda_Reichl" title="Linda Reichl">Reichl, Linda E.</a> (2004), The transition to chaos: conservative classical systems and quantum manifestations, Berlin, DE; New York, NY: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98788-0" title="Special:BookSources/978-0-387-98788-0"><bdi>978-0-387-98788-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+transition+to+chaos%3A+conservative+classical+systems+and+quantum+manifestations&rft.place=Berlin%2C+DE%3B+New+York%2C+NY&rft.pub=Springer-Verlag&rft.date=2004&rft.isbn=978-0-387-98788-0&rft.aulast=Reichl&rft.aufirst=Linda+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRowen2008" class="citation cs2">Rowen, Louis Halle (2008), Graduate Algebra: noncommutative view, Providence, RI: <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/978-0-8218-4153-2" title="Special:BookSources/978-0-8218-4153-2"><bdi>978-0-8218-4153-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Graduate+Algebra%3A+noncommutative+view&rft.place=Providence%2C+RI&rft.pub=American+Mathematical+Society&rft.date=2008&rft.isbn=978-0-8218-4153-2&rft.aulast=Rowen&rft.aufirst=Louis+Halle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFŠolin2005" class="citation cs2">Šolin, Pavel (2005), Partial Differential Equations and the Finite Element Method, <a href="/wiki/Wiley-Interscience" class="mw-redirect" title="Wiley-Interscience">Wiley-Interscience</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-76409-0" title="Special:BookSources/978-0-471-76409-0"><bdi>978-0-471-76409-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+Differential+Equations+and+the+Finite+Element+Method&rft.pub=Wiley-Interscience&rft.date=2005&rft.isbn=978-0-471-76409-0&rft.aulast=%C5%A0olin&rft.aufirst=Pavel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStinson2005" class="citation cs2">Stinson, Douglas R. (2005), Cryptography, Discrete Mathematics and its Applications, Chapman & Hall/CRC, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-508-5" title="Special:BookSources/978-1-58488-508-5"><bdi>978-1-58488-508-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cryptography&rft.series=Discrete+Mathematics+and+its+Applications&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2005&rft.isbn=978-1-58488-508-5&rft.aulast=Stinson&rft.aufirst=Douglas+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStoerBulirsch2002" class="citation cs2">Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Berlin, DE; New York, NY: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95452-3" title="Special:BookSources/978-0-387-95452-3"><bdi>978-0-387-95452-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Numerical+Analysis&rft.place=Berlin%2C+DE%3B+New+York%2C+NY&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2002&rft.isbn=978-0-387-95452-3&rft.aulast=Stoer&rft.aufirst=Josef&rft.au=Bulirsch%2C+Roland&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWard1997" class="citation cs2">Ward, J. P. (1997), <a rel="nofollow" class="external text" href="https://archive.org/details/quaternionscayle0000ward">Quaternions and Cayley numbers</a>, Mathematics and its Applications, vol. 403, Dordrecht, NL: Kluwer Academic Publishers Group, <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%2F978-94-011-5768-1">10.1007/978-94-011-5768-1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-4513-8" title="Special:BookSources/978-0-7923-4513-8"><bdi>978-0-7923-4513-8</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=1458894">1458894</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quaternions+and+Cayley+numbers&rft.place=Dordrecht%2C+NL&rft.series=Mathematics+and+its+Applications&rft.pub=Kluwer+Academic+Publishers+Group&rft.date=1997&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1458894%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-94-011-5768-1&rft.isbn=978-0-7923-4513-8&rft.aulast=Ward&rft.aufirst=J.+P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquaternionscayle0000ward&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolfram2003" class="citation cs2"><a href="/wiki/Stephen_Wolfram" title="Stephen Wolfram">Wolfram, Stephen</a> (2003), The Mathematica Book (5th ed.), Champaign, IL: Wolfram Media, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-57955-022-6" title="Special:BookSources/978-1-57955-022-6"><bdi>978-1-57955-022-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematica+Book&rft.place=Champaign%2C+IL&rft.edition=5th&rft.pub=Wolfram+Media&rft.date=2003&rft.isbn=978-1-57955-022-6&rft.aulast=Wolfram&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Physics_references">Physics references</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=45" title="Edit section: Physics references" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBohm2001" class="citation cs2">Bohm, Arno (2001), Quantum Mechanics: Foundations and Applications, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95330-2" title="Special:BookSources/0-387-95330-2"><bdi>0-387-95330-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%3A+Foundations+and+Applications&rft.pub=Springer&rft.date=2001&rft.isbn=0-387-95330-2&rft.aulast=Bohm&rft.aufirst=Arno&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgessMoore2007" class="citation cs2">Burgess, Cliff; Moore, Guy (2007), The Standard Model. A Primer, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-86036-9" title="Special:BookSources/978-0-521-86036-9"><bdi>978-0-521-86036-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Standard+Model.+A+Primer&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-86036-9&rft.aulast=Burgess&rft.aufirst=Cliff&rft.au=Moore%2C+Guy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuenther1990" class="citation cs2">Guenther, Robert D. (1990), Modern Optics, John Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-60538-7" title="Special:BookSources/0-471-60538-7"><bdi>0-471-60538-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Optics&rft.pub=John+Wiley&rft.date=1990&rft.isbn=0-471-60538-7&rft.aulast=Guenther&rft.aufirst=Robert+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFItzyksonZuber1980" class="citation cs2">Itzykson, Claude; Zuber, Jean-Bernard (1980), <a rel="nofollow" class="external text" href="https://archive.org/details/quantumfieldtheo0000itzy">Quantum Field Theory</a>, McGraw–Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-032071-3" title="Special:BookSources/0-07-032071-3"><bdi>0-07-032071-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory&rft.pub=McGraw%E2%80%93Hill&rft.date=1980&rft.isbn=0-07-032071-3&rft.aulast=Itzykson&rft.aufirst=Claude&rft.au=Zuber%2C+Jean-Bernard&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumfieldtheo0000itzy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRileyHobsonBence1997" class="citation cs2">Riley, Kenneth F.; Hobson, Michael P.; Bence, Stephen J. (1997), Mathematical methods for physics and engineering, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-55506-X" title="Special:BookSources/0-521-55506-X"><bdi>0-521-55506-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+methods+for+physics+and+engineering&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=0-521-55506-X&rft.aulast=Riley&rft.aufirst=Kenneth+F.&rft.au=Hobson%2C+Michael+P.&rft.au=Bence%2C+Stephen+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchiff1968" class="citation cs2">Schiff, Leonard I. (1968), Quantum Mechanics (3rd ed.), McGraw–Hill</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.edition=3rd&rft.pub=McGraw%E2%80%93Hill&rft.date=1968&rft.aulast=Schiff&rft.aufirst=Leonard+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg1995" class="citation cs2">Weinberg, Steven (1995), <a rel="nofollow" class="external text" href="https://archive.org/details/quantumtheoryoff00stev">The Quantum Theory of Fields. Volume I: Foundations</a>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-55001-7" title="Special:BookSources/0-521-55001-7"><bdi>0-521-55001-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quantum+Theory+of+Fields.+Volume+I%3A+Foundations&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=0-521-55001-7&rft.aulast=Weinberg&rft.aufirst=Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumtheoryoff00stev&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWherrett1987" class="citation cs2">Wherrett, Brian S. (1987), Group Theory for Atoms, Molecules and Solids, Prentice–Hall International, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-365461-3" title="Special:BookSources/0-13-365461-3"><bdi>0-13-365461-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+Theory+for+Atoms%2C+Molecules+and+Solids&rft.pub=Prentice%E2%80%93Hall+International&rft.date=1987&rft.isbn=0-13-365461-3&rft.aulast=Wherrett&rft.aufirst=Brian+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZabrodinBrezinKazakovSerban2006" class="citation cs2">Zabrodin, Anton; Brezin, Édouard; Kazakov, Vladimir; Serban, Didina; Wiegmann, Paul (2006), Applications of Random Matrices in Physics (NATO Science Series II: Mathematics, Physics and Chemistry), Berlin, DE; New York, NY: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-4530-1" title="Special:BookSources/978-1-4020-4530-1"><bdi>978-1-4020-4530-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Random+Matrices+in+Physics+%28NATO+Science+Series+II%3A+Mathematics%2C+Physics+and+Chemistry%29&rft.place=Berlin%2C+DE%3B+New+York%2C+NY&rft.pub=Springer-Verlag&rft.date=2006&rft.isbn=978-1-4020-4530-1&rft.aulast=Zabrodin&rft.aufirst=Anton&rft.au=Brezin%2C+%C3%89douard&rft.au=Kazakov%2C+Vladimir&rft.au=Serban%2C+Didina&rft.au=Wiegmann%2C+Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Historical_references">Historical references</h3> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=46" title="Edit section: Historical references" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <ul><li>A. Cayley A memoir on the theory of matrices. Phil. Trans. 148 1858 17–37; Math. Papers II 475–496</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBôcher2004" class="citation cs2"><a href="/wiki/Maxime_B%C3%B4cher" title="Maxime Bôcher">Bôcher, Maxime</a> (2004), Introduction to higher algebra, New York, NY: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49570-5" title="Special:BookSources/978-0-486-49570-5"><bdi>978-0-486-49570-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+higher+algebra&rft.place=New+York%2C+NY&rft.pub=Dover+Publications&rft.date=2004&rft.isbn=978-0-486-49570-5&rft.aulast=B%C3%B4cher&rft.aufirst=Maxime&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988">, reprint of the 1907 original edition</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCayley1889" class="citation cs2 cs1-prop-long-vol"><a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley, Arthur</a> (1889), <a rel="nofollow" class="external text" href="https://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=ABS3153.0001.001;didno=ABS3153.0001.001;view=image;seq=00000140">The collected mathematical papers of Arthur Cayley</a>, vol. I (1841–1853), <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, pp. 123–126</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+collected+mathematical+papers+of+Arthur+Cayley&rft.pages=123-126&rft.pub=Cambridge+University+Press&rft.date=1889&rft.aulast=Cayley&rft.aufirst=Arthur&rft_id=https%3A%2F%2Fquod.lib.umich.edu%2Fcgi%2Ft%2Ftext%2Fpageviewer-idx%3Fc%3Dumhistmath%3Bcc%3Dumhistmath%3Brgn%3Dfull%2520text%3Bidno%3DABS3153.0001.001%3Bdidno%3DABS3153.0001.001%3Bview%3Dimage%3Bseq%3D00000140&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDieudonné1978" class="citation cs2"><a href="/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a>, ed. (1978), Abrégé d'histoire des mathématiques 1700-1900, Paris, FR: Hermann</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abr%C3%A9g%C3%A9+d%27histoire+des+math%C3%A9matiques+1700-1900&rft.place=Paris%2C+FR&rft.pub=Hermann&rft.date=1978&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHawkins1975" class="citation cs2">Hawkins, Thomas (1975), "Cauchy and the spectral theory of matrices", <a href="/wiki/Historia_Mathematica" title="Historia Mathematica">Historia Mathematica</a>, 2: 1–29, <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%2F0315-0860%2875%2990032-4">10.1016/0315-0860(75)90032-4</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/0315-0860">0315-0860</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=0469635">0469635</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Cauchy+and+the+spectral+theory+of+matrices&rft.volume=2&rft.pages=1-29&rft.date=1975&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0469635%23id-name%3DMR&rft.issn=0315-0860&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2875%2990032-4&rft.aulast=Hawkins&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnobloch1994" class="citation cs2"><a href="/wiki/Eberhard_Knobloch" title="Eberhard Knobloch">Knobloch, Eberhard</a> (1994), "From Gauss to Weierstrass: determinant theory and its historical evaluations", The intersection of history and mathematics, Science Networks Historical Studies, vol. 15, Basel, Boston, Berlin: Birkhäuser, pp. 51–66, <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=1308079">1308079</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=From+Gauss+to+Weierstrass%3A+determinant+theory+and+its+historical+evaluations&rft.btitle=The+intersection+of+history+and+mathematics&rft.place=Basel%2C+Boston%2C+Berlin&rft.series=Science+Networks+Historical+Studies&rft.pages=51-66&rft.pub=Birkh%C3%A4user&rft.date=1994&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1308079%23id-name%3DMR&rft.aulast=Knobloch&rft.aufirst=Eberhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKronecker1897" class="citation cs2"><a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Kronecker, Leopold</a> (1897), <a href="/wiki/Kurt_Hensel" title="Kurt Hensel">Hensel, Kurt</a> (ed.), <a rel="nofollow" class="external text" href="https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAS8260.0002.001">Leopold Kronecker's Werke</a>, Teubner</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Leopold+Kronecker%27s+Werke&rft.pub=Teubner&rft.date=1897&rft.aulast=Kronecker&rft.aufirst=Leopold&rft_id=https%3A%2F%2Fquod.lib.umich.edu%2Fcgi%2Ft%2Ftext%2Ftext-idx%3Fc%3Dumhistmath%3Bidno%3DAAS8260.0002.001&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMehraRechenberg1987" class="citation cs2"><a href="/wiki/Jagdish_Mehra" title="Jagdish Mehra">Mehra, Jagdish</a>; <a href="/wiki/Helmut_Rechenberg" title="Helmut Rechenberg">Rechenberg, Helmut</a> (1987), The Historical Development of Quantum Theory (1st ed.), Berlin, DE; New York, NY: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96284-9" title="Special:BookSources/978-0-387-96284-9"><bdi>978-0-387-96284-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Historical+Development+of+Quantum+Theory&rft.place=Berlin%2C+DE%3B+New+York%2C+NY&rft.edition=1st&rft.pub=Springer-Verlag&rft.date=1987&rft.isbn=978-0-387-96284-9&rft.aulast=Mehra&rft.aufirst=Jagdish&rft.au=Rechenberg%2C+Helmut&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShenCrossleyLun1999" class="citation cs2">Shen, Kangshen; Crossley, John N.; Lun, Anthony Wah-Cheung (1999), Nine Chapters of the Mathematical Art, Companion and Commentary (2nd ed.), <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853936-0" title="Special:BookSources/978-0-19-853936-0"><bdi>978-0-19-853936-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nine+Chapters+of+the+Mathematical+Art%2C+Companion+and+Commentary&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=1999&rft.isbn=978-0-19-853936-0&rft.aulast=Shen&rft.aufirst=Kangshen&rft.au=Crossley%2C+John+N.&rft.au=Lun%2C+Anthony+Wah-Cheung&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeierstrass1915" class="citation cs2"><a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass, Karl</a> (1915), <a rel="nofollow" class="external text" href="https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAN8481.0003.001">Collected works</a>, vol. 3</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collected+works&rft.date=1915&rft.aulast=Weierstrass&rft.aufirst=Karl&rft_id=https%3A%2F%2Fquod.lib.umich.edu%2Fcgi%2Ft%2Ftext%2Ftext-idx%3Fc%3Dumhistmath%3Bidno%3DAAN8481.0003.001&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(15)"><h2 id="Further_reading">Further reading</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=47" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-15 collapsible-block" id="mf-section-15"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Matrix">"Matrix"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Matrix&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DMatrix&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://math.uwaterloo.ca/~hwolkowi//matrixcookbook.pdf">The Matrix Cookbook</a> (PDF), retrieved 24 March 2014</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_id=https%3A%2F%2Fmath.uwaterloo.ca%2F~hwolkowi%2F%2Fmatrixcookbook.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrookes2005" class="citation cs2">Brookes, Mike (2005), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081216124433/http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html">The Matrix Reference Manual</a>, London: <a href="/wiki/Imperial_College" class="mw-redirect" title="Imperial College">Imperial College</a>, archived from <a rel="nofollow" class="external text" href="https://ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html">the original</a> on 16 December 2008, retrieved 10 Dec 2008</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Matrix+Reference+Manual&rft.place=London&rft.pub=Imperial+College&rft.date=2005&rft.aulast=Brookes&rft.aufirst=Mike&rft_id=https%3A%2F%2Fee.ic.ac.uk%2Fhp%2Fstaff%2Fdmb%2Fmatrix%2Fintro.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+%28mathematics%29" class="Z3988"></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(16)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=Matrix_(mathematics)&action=edit&section=48" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-16 collapsible-block" id="mf-section-16"> <style 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href="https://en.wikibooks.org/wiki/Linear_Algebra" class="extiw" title="b:Linear Algebra">Textbooks</a> from Wikibooks</li><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png" decoding="async" width="27" height="22" class="mw-file-element" data-file-width="626" data-file-height="512"></noscript><span class="lazy-image-placeholder" style="width: 27px;height: 22px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png" data-alt="" data-width="27" data-height="22" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/41px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/54px-Wikiversity_logo_2017.svg.png 2x" data-class="mw-file-element"> <a href="https://en.wikiversity.org/wiki/Linear_algebra#Matrices" class="extiw" title="v:Linear algebra">Resources</a> from Wikiversity</li><li><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/27px-Wikidata-logo.svg.png" decoding="async" width="27" height="15" class="mw-file-element" data-file-width="1050" data-file-height="590"></noscript><span class="lazy-image-placeholder" style="width: 27px;height: 15px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/27px-Wikidata-logo.svg.png" data-alt="" data-width="27" data-height="15" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/41px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/54px-Wikidata-logo.svg.png 2x" data-class="mw-file-element"> <a href="https://www.wikidata.org/wiki/Q44337" class="extiw" title="d:Q44337">Data</a> from Wikidata</li></ul></div></div> </div> <ul><li><a rel="nofollow" class="external text" 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href="https://af.wikipedia.org/wiki/Matriks" title="Matriks – Afrikaans" lang="af" hreflang="af" data-title="Matriks" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%9B%E1%89%B5%E1%88%AA%E1%8A%AD%E1%88%B5" title="ማትሪክስ – Amharic" lang="am" hreflang="am" data-title="ማትሪክስ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مصفوفة (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="مصفوفة (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Matris" title="Matris – Azerbaijani" lang="az" hreflang="az" data-title="Matris" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%9F%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A7%8D%E0%A6%B8" title="ম্যাট্রিক্স – Bangla" lang="bn" hreflang="bn" data-title="ম্যাট্রিক্স" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/H%C3%A2ng-lia%CC%8Dt" title="Hâng-lia̍t – Minnan" lang="nan" hreflang="nan" data-title="Hâng-lia̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Матрыца (матэматыка) – Belarusian" lang="be" hreflang="be" data-title="Матрыца (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0" title="Матрыца – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матрыца" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – Bulgarian" lang="bg" hreflang="bg" data-title="Матрица (математика)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) – Bosnian" lang="bs" hreflang="bs" data-title="Matrica (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matriu_(matem%C3%A0tiques)" title="Matriu (matemàtiques) – Catalan" lang="ca" hreflang="ca" data-title="Matriu (matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Матрица (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matice" title="Matice – Czech" lang="cs" hreflang="cs" data-title="Matice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Matrics" title="Matrics – Welsh" lang="cy" hreflang="cy" data-title="Matrics" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Matrix" title="Matrix – Danish" lang="da" hreflang="da" data-title="Matrix" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%D9%8A%D8%B3" title="ماتريس – Moroccan Arabic" lang="ary" hreflang="ary" data-title="ماتريس" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Matrix_(Mathematik)" title="Matrix (Mathematik) – German" lang="de" hreflang="de" data-title="Matrix (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Maatriks" title="Maatriks – Estonian" lang="et" hreflang="et" data-title="Maatriks" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Πίνακας (μαθηματικά) – Greek" lang="el" hreflang="el" data-title="Πίνακας (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matriz_(matem%C3%A1tica)" title="Matriz (matemática) – Spanish" lang="es" hreflang="es" data-title="Matriz (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Matrico" title="Matrico – Esperanto" lang="eo" hreflang="eo" data-title="Matrico" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Matrize" title="Matrize – Basque" lang="eu" hreflang="eu" data-title="Matrize" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3" title="ماتریس – Persian" lang="fa" hreflang="fa" data-title="ماتریس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques) – French" lang="fr" hreflang="fr" data-title="Matrice (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Maitr%C3%ADs" title="Maitrís – Irish" lang="ga" hreflang="ga" data-title="Maitrís" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Matriz_(matem%C3%A1ticas)" title="Matriz (matemáticas) – Galician" lang="gl" hreflang="gl" data-title="Matriz (matemáticas)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E8%A1%8C%E5%88%97" title="行列 – Gan" lang="gan" hreflang="gan" data-title="行列" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%96%89%EB%A0%AC" title="행렬 – Korean" lang="ko" hreflang="ko" data-title="행렬" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%BF%D6%80%D5%AB%D6%81" title="Մատրից – Armenian" lang="hy" hreflang="hy" data-title="Մատրից" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%B5%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%B9" title="आव्यूह – Hindi" lang="hi" hreflang="hi" data-title="आव्यूह" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) – Croatian" lang="hr" hreflang="hr" data-title="Matrica (matematika)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matriks_(matematika)" title="Matriks (matematika) – Indonesian" lang="id" hreflang="id" data-title="Matriks (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Matrice_(mathematica)" title="Matrice (mathematica) – Interlingua" lang="ia" hreflang="ia" data-title="Matrice (mathematica)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fylki_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Fylki (stærðfræði) – Icelandic" lang="is" hreflang="is" data-title="Fylki (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matrice" title="Matrice – Italian" lang="it" hreflang="it" data-title="Matrice" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94" title="מטריצה – Hebrew" lang="he" hreflang="he" data-title="מטריצה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AE%E0%B2%BE%E0%B2%A4%E0%B3%83%E0%B2%95%E0%B3%86%E0%B2%97%E0%B2%B3%E0%B3%81" title="ಮಾತೃಕೆಗಳು – Kannada" lang="kn" hreflang="kn" data-title="ಮಾತೃಕೆಗಳು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%A2%E1%83%A0%E1%83%98%E1%83%AA%E1%83%90_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="მატრიცა (მათემატიკა) – Georgian" lang="ka" hreflang="ka" data-title="მატრიცა (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – Kazakh" lang="kk" hreflang="kk" data-title="Матрица (математика)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Matris_(mat%C3%A9matik)" title="Matris (matématik) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Matris (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%A1%E0%BA%B2%E0%BA%95%E0%BA%A3%E0%BA%B4%E0%BA%81" title="ມາຕຣິກ – Lao" lang="lo" hreflang="lo" data-title="ມາຕຣິກ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Matrix_(mathematica)" title="Matrix (mathematica) – Latin" lang="la" hreflang="la" data-title="Matrix (mathematica)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Matrica" title="Matrica – Latvian" lang="lv" hreflang="lv" data-title="Matrica" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) – Lithuanian" lang="lt" hreflang="lt" data-title="Matrica (matematika)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Matris" title="Matris – Lombard" lang="lmo" hreflang="lmo" data-title="Matris" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/M%C3%A1trix_(matematika)" title="Mátrix (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Mátrix (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – Macedonian" lang="mk" hreflang="mk" data-title="Матрица (математика)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AE%E0%B4%BE%E0%B4%9F%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B5%8D%E0%B4%B8%E0%B5%8D" title="മാട്രിക്സ് – Malayalam" lang="ml" hreflang="ml" data-title="മാട്രിക്സ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Matriks_(matematik)" title="Matriks (matematik) – Malay" lang="ms" hreflang="ms" data-title="Matriks (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%A1%E1%80%AF%E1%80%B6" title="ကိန်းအုံ – Burmese" lang="my" hreflang="my" data-title="ကိန်းအုံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Matrix_(wiskunde)" title="Matrix (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Matrix (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AE%E0%A5%87%E0%A4%9F%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8" title="मेट्रिक्स – Nepali" lang="ne" hreflang="ne" data-title="मेट्रिक्स" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A1%8C%E5%88%97" title="行列 – Japanese" lang="ja" hreflang="ja" data-title="行列" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Maatriks" title="Maatriks – Northern Frisian" lang="frr" hreflang="frr" data-title="Maatriks" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matrise" title="Matrise – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Matrise" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matrise" title="Matrise – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matrise" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B5" title="Матрице – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Матрице" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target">Олык марий</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AE%E0%AC%BE%E0%AC%9F%E0%AD%8D%E0%AC%B0%E0%AC%BF%E0%AC%95%E0%AD%8D%E0%AC%B8" title="ମାଟ୍ରିକ୍ସ – Odia" lang="or" hreflang="or" data-title="ମାଟ୍ରିକ୍ସ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Tareentaa_(Maatiriksii)" title="Tareentaa (Maatiriksii) – Oromo" lang="om" hreflang="om" data-title="Tareentaa (Maatiriksii)" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matritsa_matematikada" title="Matritsa matematikada – Uzbek" lang="uz" hreflang="uz" data-title="Matritsa matematikada" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AE%E0%A9%88%E0%A8%9F%E0%A9%8D%E0%A8%B0%E0%A8%BF%E0%A8%95%E0%A8%B8_(%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4)" title="ਮੈਟ੍ਰਿਕਸ (ਗਣਿਤ) – Punjabi" lang="pa" hreflang="pa" data-title="ਮੈਟ੍ਰਿਕਸ (ਗਣਿਤ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D8%A7%D9%B9%D8%B1%DA%A9%D8%B3_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="ماٹرکس (ریاضیات) – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ماٹرکس (ریاضیات)" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Matris" title="Matris – Piedmontese" lang="pms" hreflang="pms" data-title="Matris" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Macierz" title="Macierz – Polish" lang="pl" hreflang="pl" data-title="Macierz" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matriz_(matem%C3%A1tica)" title="Matriz (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Matriz (matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Matrice" title="Matrice – Romanian" lang="ro" hreflang="ro" data-title="Matrice" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – Russian" lang="ru" hreflang="ru" data-title="Матрица (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – Yakut" lang="sah" hreflang="sah" data-title="Матрица (математика)" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics) – Scots" lang="sco" hreflang="sco" data-title="Matrix (mathematics)" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matrica" title="Matrica – Albanian" lang="sq" hreflang="sq" data-title="Matrica" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Matrici_(matim%C3%A0tica)" title="Matrici (matimàtica) – Sicilian" lang="scn" hreflang="scn" data-title="Matrici (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://si.wikipedia.org/wiki/%E0%B6%B1%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%83_(%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%BA)" title="න්‍යාස (ගණිතය) – Sinhala" lang="si" hreflang="si" data-title="න්‍යාස (ගණිතය)" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Matrix (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matica_(matematika)" title="Matica (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Matica (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matrika" title="Matrika – Slovenian" lang="sl" hreflang="sl" data-title="Matrika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Taxane" title="Taxane – Somali" lang="so" hreflang="so" data-title="Taxane" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%DA%A9%D8%B3" title="ماتریکس – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ماتریکس" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" 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Matrix (mathematics) - Wikipedia