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<div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Specific patterns for entries</span> </div> </a> <ul id="toc-Specific_patterns_for_entries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrices_satisfying_some_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrices_satisfying_some_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Matrices satisfying some equations</span> </div> </a> <ul id="toc-Matrices_satisfying_some_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrices_with_conditions_on_eigenvalues_or_eigenvectors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrices_with_conditions_on_eigenvalues_or_eigenvectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Matrices with conditions on eigenvalues or eigenvectors</span> </div> </a> <ul id="toc-Matrices_with_conditions_on_eigenvalues_or_eigenvectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrices_generated_by_specific_data" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrices_generated_by_specific_data"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Matrices generated by specific data</span> </div> </a> <ul id="toc-Matrices_generated_by_specific_data-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrices_used_in_statistics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrices_used_in_statistics"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Matrices used in statistics</span> </div> </a> <ul id="toc-Matrices_used_in_statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrices_used_in_graph_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrices_used_in_graph_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Matrices used in graph theory</span> </div> </a> <ul id="toc-Matrices_used_in_graph_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrices_used_in_science_and_engineering" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrices_used_in_science_and_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Matrices used in science and engineering</span> </div> </a> <ul id="toc-Matrices_used_in_science_and_engineering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specific_matrices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Specific_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Specific matrices</span> </div> </a> <ul id="toc-Specific_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_matrix-related_terms_and_definitions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_matrix-related_terms_and_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Other matrix-related terms and definitions</span> </div> </a> <ul id="toc-Other_matrix-related_terms_and_definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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</ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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href="/wiki/File:Taxonomy_of_Complex_Matrices.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Taxonomy_of_Complex_Matrices.svg/247px-Taxonomy_of_Complex_Matrices.svg.png" decoding="async" width="247" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Taxonomy_of_Complex_Matrices.svg/371px-Taxonomy_of_Complex_Matrices.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Taxonomy_of_Complex_Matrices.svg/494px-Taxonomy_of_Complex_Matrices.svg.png 2x" data-file-width="1350" data-file-height="900" /></a><figcaption>Several important classes of matrices are subsets of each other.</figcaption></figure> <p>This article lists some important classes of <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> used in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Science" title="Science">science</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>. A <b>matrix</b> (plural matrices, or less commonly matrixes) is a rectangular <a href="/wiki/Array_data_structure" class="mw-redirect" title="Array data structure">array</a> of <a href="/wiki/Number" title="Number">numbers</a> called <i>entries</i>. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. Important examples include the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10983f1fccdf00b7095eea8457480727538b91ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:23.273ex; height:14.176ex;" alt="{\displaystyle I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}.}"></span></dd></dl> <p>and the <a href="/wiki/Zero_matrix" title="Zero matrix">zero matrix</a> of dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>. For example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{2\times 3}={\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>×</mo> <mn>3</mn> </mrow> </msub> <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>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O_{2\times 3}={\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411eb6114fb4bc1924cf6ddbe3ce0fbaded49fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.332ex; height:6.176ex;" alt="{\displaystyle O_{2\times 3}={\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}}}"></span>.</dd></dl> <p>Further ways of classifying matrices are according to their <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>, or by imposing conditions on the <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">product</a> of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences including <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, have particular matrices that are applied chiefly in these areas. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Constant_matrices">Constant matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=1" title="Edit section: Constant matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The list below comprises matrices whose elements are constant for any given dimension (size) of matrix. The matrix entries will be denoted <i>a<sub>ij</sub></i>. The table below uses the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a> δ<sub><i>ij</i></sub> for two integers <i>i</i> and <i>j</i> which is 1 if <i>i</i> = <i>j</i> and 0 else. </p> <table class="wikitable sortable"> <tbody><tr> <th>Name</th> <th>Explanation</th> <th>Symbolic description of the entries</th> <th>Notes </th></tr> <tr> <td><a href="/wiki/Commutation_matrix" title="Commutation matrix">Commutation matrix</a></td> <td>The matrix of the <a href="/wiki/Linear_map" title="Linear map">linear map</a> that maps a matrix to its transpose</td> <td></td> <td>See <a href="/wiki/Vectorization_(mathematics)" title="Vectorization (mathematics)">Vectorization</a> </td></tr> <tr> <td><a href="/wiki/Duplication_matrix" class="mw-redirect" title="Duplication matrix">Duplication matrix</a></td> <td>The matrix of the linear map mapping the vector of the distinct entries of a <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a> to the vector of all entries of the matrix</td> <td></td> <td>See <a href="/wiki/Vectorization_(mathematics)" title="Vectorization (mathematics)">Vectorization</a> </td></tr> <tr> <td><a href="/wiki/Elimination_matrix" class="mw-redirect" title="Elimination matrix">Elimination matrix</a></td> <td>The matrix of the linear map mapping the vector of the entries of a matrix to the vector of a part of the entries (for example the vector of the entries that are not below the main diagonal)</td> <td></td> <td>See <a href="/wiki/Vectorization_(mathematics)" title="Vectorization (mathematics)">vectorization</a> </td></tr> <tr> <td><a href="/wiki/Exchange_matrix" title="Exchange matrix">Exchange matrix</a></td> <td>The <a href="/wiki/Binary_matrix" class="mw-redirect" title="Binary matrix">binary matrix</a> with ones on the anti-diagonal, and zeroes everywhere else.</td> <td><i>a<sub>ij</sub></i> = δ<sub><i>n</i>+1−<i>i</i>,<i>j</i></sub></td> <td>A <a href="/wiki/Permutation_matrix" title="Permutation matrix">permutation matrix</a>. </td></tr> <tr> <td><a href="/wiki/Hilbert_matrix" title="Hilbert matrix">Hilbert matrix</a></td> <td></td> <td><i>a</i><sub><i>ij</i></sub> = (<i>i</i> + <i>j</i> − 1)<sup>−1</sup>.</td> <td>A <a href="/wiki/Hankel_matrix" title="Hankel matrix">Hankel matrix</a>. </td></tr> <tr> <td><a href="/wiki/Identity_matrix" title="Identity matrix">Identity matrix</a></td> <td>A square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0.</td> <td><i>a<sub>ij</sub></i> = δ<sub><i>ij</i></sub></td> <td> </td></tr> <tr> <td><a href="/wiki/Lehmer_matrix" title="Lehmer matrix">Lehmer matrix</a></td> <td></td> <td><i>a<sub>ij</sub></i> = min(<i>i</i>, <i>j</i>) ÷ max(<i>i</i>, <i>j</i>).</td> <td>A <a href="/wiki/Positive_matrix" class="mw-redirect" title="Positive matrix">positive</a> <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a>. </td></tr> <tr> <td><a href="/wiki/Matrix_of_ones" title="Matrix of ones">Matrix of ones</a></td> <td>A matrix with all entries equal to one.</td> <td><i>a<sub>ij</sub></i> = 1.</td> <td> </td></tr> <tr> <td><a href="/wiki/Pascal_matrix" title="Pascal matrix">Pascal matrix</a></td> <td>A matrix containing the entries of <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>.</td> <td></td> <td> </td></tr> <tr> <td><a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a></td> <td>A set of three 2 × 2 complex Hermitian and unitary matrices. When combined with the <i>I</i><sub>2</sub> identity matrix, they form an orthogonal basis for the 2 × 2 complex Hermitian matrices.</td> <td></td> <td> </td></tr> <tr> <td><a href="/wiki/Redheffer_matrix" title="Redheffer matrix">Redheffer matrix</a></td> <td>Encodes a <a href="/wiki/Dirichlet_convolution" title="Dirichlet convolution">Dirichlet convolution</a>. Matrix entries are given by the <a href="/wiki/Divisor_function" title="Divisor function">divisor function</a>; entires of the inverse are given by the <a href="/wiki/M%C3%B6bius_function" title="Möbius function">Möbius function</a>.</td> <td><i>a</i><sub><i>ij</i></sub> are 1 if <i>i</i> divides <i>j</i> or if <i>j</i> = 1; otherwise, <i>a</i><sub><i>ij</i></sub> = 0.</td> <td>A (0, 1)-matrix. </td></tr> <tr> <td><a href="/wiki/Shift_matrix" title="Shift matrix">Shift matrix</a></td> <td>A matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere.</td> <td><i>a<sub>ij</sub></i> = δ<sub><i>i</i>+1,<i>j</i></sub> or <i>a<sub>ij</sub></i> = δ<sub><i>i</i>−1,<i>j</i></sub></td> <td>Multiplication by it shifts matrix elements by one position. </td></tr> <tr> <td><a href="/wiki/Zero_matrix" title="Zero matrix">Zero matrix</a></td> <td>A matrix with all entries equal to zero.</td> <td><i>a<sub>ij</sub></i> = 0.</td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Specific_patterns_for_entries">Specific patterns for entries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=2" title="Edit section: Specific patterns for entries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following lists matrices whose entries are subject to certain conditions. Many of them apply to <i>square matrices</i> only, that is matrices with the same number of columns and rows. The <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> of a square matrix is the <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> joining the upper left corner and the lower right one or equivalently the entries <i>a</i><sub><i>i</i>,<i>i</i></sub>. The other diagonal is called anti-diagonal (or counter-diagonal). </p> <table class="wikitable sortable"> <tbody><tr> <th>Name</th> <th>Explanation</th> <th>Notes, references </th></tr> <tr> <td><a href="/wiki/Logical_matrix" title="Logical matrix">(0,1)-matrix</a></td> <td>A matrix with all elements either 0 or 1.</td> <td>Synonym for <i>binary matrix</i> or <i>logical matrix</i>. </td></tr> <tr> <td><a href="/wiki/Alternant_matrix" title="Alternant matrix">Alternant matrix</a></td> <td>A matrix in which successive columns have a particular function applied to their entries.</td> <td> </td></tr> <tr> <td><a href="/wiki/Alternating_sign_matrix" title="Alternating sign matrix">Alternating sign matrix</a></td> <td>A square matrix with entries 0, 1 and −1 such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign.</td> <td> </td></tr> <tr> <td><a href="/wiki/Anti-diagonal_matrix" title="Anti-diagonal matrix">Anti-diagonal matrix</a></td> <td>A square matrix with all entries off the anti-diagonal equal to zero.</td> <td> </td></tr> <tr> <td><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Anti-Hermitian matrix</a></td> <td></td> <td>Synonym for <i>skew-Hermitian matrix</i>. </td></tr> <tr> <td><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Anti-symmetric matrix</a></td> <td></td> <td>Synonym for <i>skew-symmetric matrix</i>. </td></tr> <tr> <td><a href="/wiki/Arrowhead_matrix" title="Arrowhead matrix">Arrowhead matrix</a></td> <td>A square matrix containing zeros in all entries except for the first row, first column, and main diagonal.</td> <td> </td></tr> <tr> <td><a href="/wiki/Band_matrix" title="Band matrix">Band matrix</a></td> <td>A square matrix whose non-zero entries are confined to a diagonal <i>band</i>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Bidiagonal_matrix" title="Bidiagonal matrix">Bidiagonal matrix</a></td> <td>A matrix with elements only on the main diagonal and either the superdiagonal or subdiagonal.</td> <td>Sometimes defined differently, see article. </td></tr> <tr> <td><a href="/wiki/Logical_matrix" title="Logical matrix">Binary matrix</a></td> <td>A matrix whose entries are all either 0 or 1.</td> <td>Synonym for <i>(0,1)-matrix</i> or <i>logical matrix</i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Bisymmetric_matrix" title="Bisymmetric matrix">Bisymmetric matrix</a></td> <td>A square matrix that is symmetric with respect to its main diagonal and its main cross-diagonal.</td> <td> </td></tr> <tr> <td><a href="/wiki/Block-diagonal_matrix" class="mw-redirect" title="Block-diagonal matrix">Block-diagonal matrix</a></td> <td>A <a href="/wiki/Block_matrix" title="Block matrix">block matrix</a> with entries only on the diagonal.</td> <td> </td></tr> <tr> <td><a href="/wiki/Block_matrix" title="Block matrix">Block matrix</a></td> <td>A matrix partitioned in sub-matrices called blocks.</td> <td> </td></tr> <tr> <td><a href="/wiki/Block_tridiagonal_matrix" class="mw-redirect" title="Block tridiagonal matrix">Block tridiagonal matrix</a></td> <td>A block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements.</td> <td> </td></tr> <tr> <td><a href="/wiki/Boolean_matrix" title="Boolean matrix">Boolean matrix</a></td> <td>A matrix whose entries are taken from a <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Cauchy_matrix" title="Cauchy matrix">Cauchy matrix</a></td> <td>A matrix whose elements are of the form 1/(<i>x<sub>i</sub></i> + <i>y<sub>j</sub></i>) for (<i>x<sub>i</sub></i>), (<i>y<sub>j</sub></i>) injective sequences (i.e., taking every value only once).</td> <td> </td></tr> <tr> <td><a href="/wiki/Centrosymmetric_matrix" title="Centrosymmetric matrix">Centrosymmetric matrix</a></td> <td>A matrix symmetric about its center; i.e., <i>a</i><sub><i>ij</i></sub> = <i>a</i><sub><i>n</i>−<i>i</i>+1,<i>n</i>−<i>j</i>+1</sub>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Circulant_matrix" title="Circulant matrix">Circulant matrix</a></td> <td>A matrix where each row is a circular shift of its predecessor.</td> <td> </td></tr> <tr> <td><a href="/wiki/Conference_matrix" title="Conference matrix">Conference matrix</a></td> <td>A square matrix with zero diagonal and +1 and −1 off the diagonal, such that C<sup>T</sup>C is a multiple of the identity matrix.</td> <td> </td></tr> <tr> <td><a href="/wiki/Complex_Hadamard_matrix" title="Complex Hadamard matrix">Complex Hadamard matrix</a></td> <td>A matrix with all rows and columns mutually orthogonal, whose entries are unimodular.</td> <td> </td></tr> <tr> <td><a href="/wiki/Compound_matrix" title="Compound matrix">Compound matrix</a> </td> <td>A matrix whose entries are generated by the determinants of all minors of a matrix. </td> <td> </td></tr> <tr> <td><a href="/wiki/Copositive_matrix" title="Copositive matrix">Copositive matrix</a></td> <td>A square matrix <i>A</i> with real coefficients, such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{T}Ax}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>A</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x^{T}Ax}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfdfeb624d92a4f3d0d3e0b4a2135a67cfb66e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.308ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{T}Ax}"></span> is nonnegative for every nonnegative vector <i>x</i></td> <td> </td></tr> <tr> <td><a href="/wiki/Diagonally_dominant_matrix" title="Diagonally dominant matrix">Diagonally dominant matrix</a></td> <td>A matrix whose entries satisfy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a_{ii}|>\sum _{j\neq i}|a_{ij}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{ii}|>\sum _{j\neq i}|a_{ij}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8754ac4f277f3a3f36f1a23178a492dcd56955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:14.732ex; height:6.009ex;" alt="{\displaystyle |a_{ii}|>\sum _{j\neq i}|a_{ij}|}"></span>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Diagonal_matrix" title="Diagonal matrix">Diagonal matrix</a></td> <td>A square matrix with all entries outside the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> equal to zero.</td> <td> </td></tr> <tr> <td><a href="/wiki/DFT_matrix" title="DFT matrix">Discrete Fourier-transform matrix</a></td> <td>Multiplying by a vector gives the DFT of the vector as result.</td> <td> </td></tr> <tr> <td><a href="/wiki/Elementary_matrix" title="Elementary matrix">Elementary matrix</a></td> <td>A square matrix derived by applying an elementary row operation to the identity matrix.</td> <td> </td></tr> <tr> <td><a href="/wiki/Equivalent_matrix" class="mw-redirect" title="Equivalent matrix">Equivalent matrix</a></td> <td>A matrix that can be derived from another matrix through a sequence of elementary row or column operations.</td> <td> </td></tr> <tr> <td><a href="/wiki/Frobenius_matrix" title="Frobenius matrix">Frobenius matrix</a></td> <td>A square matrix in the form of an identity matrix but with arbitrary entries in one column below the main diagonal.</td> <td> </td></tr> <tr> <td><a href="/wiki/GCD_matrix" title="GCD matrix">GCD matrix</a></td> <td>The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7fcd27e8d01fdf5fe00da4f97045f079cd97bff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.309ex; height:2.843ex;" alt="{\displaystyle (S)}"></span> having the greatest common divisor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i},x_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{i},x_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/310cfb18dea1a04a6f655b0131a04cf2527908b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.212ex; height:3.009ex;" alt="{\displaystyle (x_{i},x_{j})}"></span> as its <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ij}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ij}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53fcc7b57da64979c370eb150eb5a61a625a08e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.761ex; height:2.509ex;" alt="{\displaystyle ij}"></span> entry, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i},x_{j}\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i},x_{j}\in S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a0b8b8e919d8c4694e2a6cef64c26ff84889a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.743ex; height:2.843ex;" alt="{\displaystyle x_{i},x_{j}\in S}"></span>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Generalized_permutation_matrix" title="Generalized permutation matrix">Generalized permutation matrix</a></td> <td>A square matrix with precisely one nonzero element in each row and column.</td> <td> </td></tr> <tr> <td><a href="/wiki/Hadamard_matrix" title="Hadamard matrix">Hadamard matrix</a></td> <td>A square matrix with entries +1, −1 whose rows are mutually orthogonal.</td> <td> </td></tr> <tr> <td><a href="/wiki/Hankel_matrix" title="Hankel matrix">Hankel matrix</a></td> <td>A matrix with constant skew-diagonals; also an upside down Toeplitz matrix.</td> <td>A square Hankel matrix is symmetric. </td></tr> <tr> <td><a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrix</a></td> <td>A square matrix which is equal to its <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>, <i>A</i> = <i>A</i><sup>*</sup>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg matrix</a></td> <td>An "almost" triangular matrix, for example, an upper Hessenberg matrix has zero entries below the first subdiagonal.</td> <td> </td></tr> <tr> <td><a href="/wiki/Hollow_matrix" title="Hollow matrix">Hollow matrix</a></td> <td>A square matrix whose main diagonal comprises only zero elements.</td> <td> </td></tr> <tr> <td><a href="/wiki/Integer_matrix" title="Integer matrix">Integer matrix</a></td> <td>A matrix whose entries are all integers.</td> <td> </td></tr> <tr> <td><a href="/wiki/Logical_matrix" title="Logical matrix">Logical matrix</a></td> <td>A matrix with all entries either 0 or 1.</td> <td>Synonym for <i>(0,1)-matrix</i>, <i>binary matrix</i> or <i>Boolean matrix</i>. Can be used to represent a <i>k</i>-adic <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a>. </td></tr> <tr> <td><a href="/wiki/Markov_matrix" class="mw-redirect" title="Markov matrix">Markov matrix</a></td> <td>A matrix of non-negative real numbers, such that the entries in each row sum to 1.</td> <td> </td></tr> <tr> <td><a href="/wiki/Metzler_matrix" title="Metzler matrix">Metzler matrix</a></td> <td>A matrix whose off-diagonal entries are non-negative.</td> <td> </td></tr> <tr> <td><a href="/wiki/Generalized_permutation_matrix" title="Generalized permutation matrix">Monomial matrix</a></td> <td>A square matrix with exactly one non-zero entry in each row and column.</td> <td>Synonym for <i>generalized permutation matrix</i>. </td></tr> <tr> <td><a href="/wiki/Moore_matrix" title="Moore matrix">Moore matrix</a></td> <td>A row consists of <i>a</i>, <i>a</i><sup><i>q</i></sup>, <i>a</i><sup><i>q</i>²</sup>, etc., and each row uses a different variable.</td> <td> </td></tr> <tr> <td><a href="/wiki/Nonnegative_matrix" title="Nonnegative matrix">Nonnegative matrix</a></td> <td>A matrix with all nonnegative entries.</td> <td> </td></tr> <tr> <td>Null-symmetric matrix </td> <td>A square matrix whose null space (or <a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">kernel</a>) is equal to its <a href="/wiki/Transpose" title="Transpose">transpose</a>, N(<i>A)</i> = N(<i>A<sup>T</sup></i>) or ker(<i>A)</i> = ker(<i>A<sup>T</sup></i>). </td> <td>Synonym for kernel-symmetric matrices. Examples include (but not limited to) symmetric, skew-symmetric, and normal matrices. </td></tr> <tr> <td>Null-Hermitian matrix </td> <td>A square matrix whose null space (or <a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">kernel</a>) is equal to its <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>, N(<i>A</i>)=N(<i>A</i><sup>*</sup>) or ker(<i>A</i>)=ker(<i>A</i><sup>*</sup>). </td> <td>Synonym for kernel-Hermitian matrices. Examples include (but not limited) to Hermitian, skew-Hermitian matrices, and normal matrices. </td></tr> <tr> <td><a href="/wiki/Block_matrix" title="Block matrix">Partitioned matrix</a></td> <td>A matrix partitioned into sub-matrices, or equivalently, a matrix whose entries are themselves matrices rather than scalars.</td> <td>Synonym for <i>block matrix</i>. </td></tr> <tr> <td><a href="/w/index.php?title=Parisi_matrix&action=edit&redlink=1" class="new" title="Parisi matrix (page does not exist)">Parisi matrix</a></td> <td>A block-hierarchical matrix. It consist of growing blocks placed along the diagonal, each block is itself a Parisi matrix of a smaller size.</td> <td>In theory of spin-glasses is also known as a replica matrix. </td></tr> <tr> <td><a href="/wiki/Pentadiagonal_matrix" class="mw-redirect" title="Pentadiagonal matrix">Pentadiagonal matrix</a></td> <td>A matrix with the only nonzero entries on the main diagonal and the two diagonals just above and below the main one.</td> <td> </td></tr> <tr> <td><a href="/wiki/Permutation_matrix" title="Permutation matrix">Permutation matrix</a></td> <td>A matrix representation of a <a href="/wiki/Permutation" title="Permutation">permutation</a>, a square matrix with exactly one 1 in each row and column, and all other elements 0.</td> <td> </td></tr> <tr> <td><a href="/wiki/Persymmetric_matrix" title="Persymmetric matrix">Persymmetric matrix</a></td> <td>A matrix that is symmetric about its northeast–southwest diagonal, i.e., <i>a</i><sub><i>ij</i></sub> = <i>a</i><sub><i>n</i>−<i>j</i>+1,<i>n</i>−<i>i</i>+1</sub>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Polynomial_matrix" title="Polynomial matrix">Polynomial matrix</a></td> <td>A matrix whose entries are <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Positive_matrix" class="mw-redirect" title="Positive matrix">Positive matrix</a></td> <td>A matrix with all positive entries.</td> <td> </td></tr> <tr> <td><a href="/wiki/Quaternionic_matrix" title="Quaternionic matrix">Quaternionic matrix</a></td> <td>A matrix whose entries are <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Random_matrix" title="Random matrix">Random matrix</a></td> <td>A matrix whose entries are <a href="/wiki/Random_variable" title="Random variable">random variables</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Sign_matrix" class="mw-redirect" title="Sign matrix">Sign matrix</a></td> <td>A matrix whose entries are either +1, 0, or −1.</td> <td> </td></tr> <tr> <td><a href="/wiki/Signature_matrix" title="Signature matrix">Signature matrix</a></td> <td>A diagonal matrix where the diagonal elements are either +1 or −1.</td> <td> </td></tr> <tr> <td><a href="/wiki/Single-entry_matrix" class="mw-redirect" title="Single-entry matrix">Single-entry matrix</a></td> <td>A matrix where a single element is one and the rest of the elements are zero.</td> <td> </td></tr> <tr> <td><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian matrix</a></td> <td>A square matrix which is equal to the negative of its <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>, <i>A</i><sup>*</sup> = −<i>A</i>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Skew-symmetric matrix</a></td> <td>A matrix which is equal to the negative of its <a href="/wiki/Transpose" title="Transpose">transpose</a>, <i>A</i><sup>T</sup> = −<i>A</i>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Skyline_matrix" title="Skyline matrix">Skyline matrix</a></td> <td>A rearrangement of the entries of a banded matrix which requires less space.</td> <td> </td></tr> <tr> <td><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse matrix</a></td> <td>A matrix with relatively few non-zero elements.</td> <td>Sparse matrix algorithms can tackle huge sparse matrices that are utterly impractical for dense matrix algorithms. </td></tr> <tr> <td><a href="/wiki/Symmetric_matrix" title="Symmetric matrix">Symmetric matrix</a></td> <td>A square matrix which is equal to its <a href="/wiki/Transpose" title="Transpose">transpose</a>, <i>A</i> = <i>A</i><sup>T</sup> (<i>a</i><sub><i>i</i>,<i>j</i></sub> = <i>a</i><sub><i>j</i>,<i>i</i></sub>).</td> <td> </td></tr> <tr> <td><a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz matrix</a></td> <td>A matrix with constant diagonals.</td> <td> </td></tr> <tr> <td><a href="/wiki/Totally_positive_matrix" title="Totally positive matrix">Totally positive matrix</a></td> <td>A matrix with <a href="/wiki/Determinant" title="Determinant">determinants</a> of all its square submatrices positive.</td> <td> </td></tr> <tr> <td><a href="/wiki/Triangular_matrix" title="Triangular matrix">Triangular matrix</a></td> <td>A matrix with all entries above the main diagonal equal to zero (lower triangular) or with all entries below the main diagonal equal to zero (upper triangular).</td> <td> </td></tr> <tr> <td><a href="/wiki/Tridiagonal_matrix" title="Tridiagonal matrix">Tridiagonal matrix</a></td> <td>A matrix with the only nonzero entries on the main diagonal and the diagonals just above and below the main one.</td> <td> </td></tr> <tr> <td>X–Y–Z matrix </td> <td>A generalization to three dimensions of the concept of <a href="/wiki/Two-dimensional_array" class="mw-redirect" title="Two-dimensional array">two-dimensional array</a> </td></tr> <tr> <td><a href="/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde matrix</a></td> <td>A row consists of 1, <i>a</i>, <i>a</i><sup>2</sup>, <i>a</i><sup>3</sup>, etc., and each row uses a different variable.</td> <td> </td></tr> <tr> <td><a href="/wiki/Walsh_matrix" title="Walsh matrix">Walsh matrix</a></td> <td>A square matrix, with dimensions a power of 2, the entries of which are +1 or −1, and the property that the dot product of any two distinct rows (or columns) is zero.</td> <td> </td></tr> <tr> <td><a href="/wiki/Z-matrix_(mathematics)" title="Z-matrix (mathematics)">Z-matrix</a></td> <td>A matrix with all off-diagonal entries less than zero. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Matrices_satisfying_some_equations">Matrices satisfying some equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=3" title="Edit section: Matrices satisfying some equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A number of matrix-related notions is about properties of products or inverses of the given matrix. The <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a> of a <i>m</i>-by-<i>n</i> matrix <i>A</i> and a <i>n</i>-by-<i>k</i> matrix <i>B</i> is the <i>m</i>-by-<i>k</i> matrix <i>C</i> given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (C)_{i,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> <mi>C</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</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 (C)_{i,j}=\sum _{r=1}^{n}A_{i,r}B_{r,j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd263f78d70ec11b4bc19d1856cb24d19ccd248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.612ex; height:6.843ex;" alt="{\displaystyle (C)_{i,j}=\sum _{r=1}^{n}A_{i,r}B_{r,j}.}"></span><sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <p>This matrix product is denoted <i>AB</i>. Unlike the product of numbers, matrix products are not <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, that is to say <i>AB</i> need not be equal to <i>BA</i>.<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A number of notions are concerned with the failure of this commutativity. An <a href="/wiki/Inverse_of_a_matrix" class="mw-redirect" title="Inverse of a matrix">inverse</a> of square matrix <i>A</i> is a matrix <i>B</i> (necessarily of the same dimension as <i>A</i>) such that <i>AB</i> = <i>I</i>. Equivalently, <i>BA</i> = <i>I</i>. An inverse need not exist. If it exists, <i>B</i> is uniquely determined, and is also called <i>the</i> inverse of <i>A</i>, denoted <i>A</i><sup>−1</sup>. </p> <table class="wikitable sortable"> <tbody><tr> <th>Name</th> <th>Explanation</th> <th>Notes </th></tr> <tr> <td><a href="/w/index.php?title=Circular_matrix&action=edit&redlink=1" class="new" title="Circular matrix (page does not exist)">Circular matrix</a> or <a href="/w/index.php?title=Coninvolutory_matrix&action=edit&redlink=1" class="new" title="Coninvolutory matrix (page does not exist)">Coninvolutory matrix</a> </td> <td>A matrix whose inverse is equal to its entrywise complex conjugate: <i>A</i><sup>−1</sup> = <span style="text-decoration:overline;"><i>A</i></span>. </td> <td>Compare with unitary matrices. </td></tr> <tr> <td><a href="/wiki/Matrix_congruence" title="Matrix congruence">Congruent matrix</a></td> <td>Two matrices <i>A</i> and <i>B</i> are congruent if there exists an invertible matrix <i>P</i> such that <span class="nowrap"><i>P</i><sup>T</sup> <i>A</i> <i>P</i></span> = <i>B</i>.</td> <td>Compare with similar matrices. </td></tr> <tr> <td><a href="/wiki/EP_matrix" title="EP matrix">EP matrix</a> or Range-Hermitian matrix </td> <td>A square matrix that commutes with its <a href="/wiki/Moore%E2%80%93Penrose_inverse" title="Moore–Penrose inverse">Moore–Penrose inverse</a>: <i>AA</i><sup>+</sup> = <i>A</i><sup>+</sup><i>A.</i> </td> <td> </td></tr> <tr> <td><a href="/wiki/Idempotent_matrix" title="Idempotent matrix">Idempotent matrix</a> or <br /> <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Projection Matrix</a></td> <td>A matrix that has the property <i>A</i>² = <i>AA</i> = <i>A</i>.</td> <td>The name projection matrix inspires from the observation of projection of a point multiple <br /> times onto a subspace(plane or a line) giving the same result as <a href="/wiki/Projection_(linear_algebra)#Properties_and_classification" title="Projection (linear algebra)">one projection</a>. </td></tr> <tr> <td><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible matrix</a></td> <td>A square matrix having a multiplicative <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse</a>, that is, a matrix <i>B</i> such that <i>AB</i> = <i>BA</i> = <i>I</i>.</td> <td>Invertible matrices form the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a>. </td></tr> <tr> <td><a href="/wiki/Involutory_matrix" title="Involutory matrix">Involutory matrix</a></td> <td>A square matrix which is its own inverse, i.e., <i>AA</i> = <i>I</i>.</td> <td><a href="/wiki/Signature_matrix" title="Signature matrix">Signature matrices</a>, <a href="/wiki/Householder_transformation#Definition_and_properties" title="Householder transformation">Householder matrices</a> (Also known as 'reflection matrices' <br /> to reflect a point about a plane or line) have this property. </td></tr> <tr> <td><a href="/wiki/Isometry" title="Isometry">Isometric matrix</a></td> <td>A matrix that preserves distances, i.e., a matrix that satisfies <i>A</i><sup>*</sup><i>A</i> = <i>I</i> where <i>A</i><sup>*</sup> denotes the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of <i>A</i>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Nilpotent_matrix" title="Nilpotent matrix">Nilpotent matrix</a></td> <td>A square matrix satisfying <i>A</i><sup><i>q</i></sup> = 0 for some positive integer <i>q</i>.</td> <td>Equivalently, the only eigenvalue of <i>A</i> is 0. </td></tr> <tr> <td><a href="/wiki/Normal_matrix" title="Normal matrix">Normal matrix</a></td> <td>A square matrix that commutes with its <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>: <i>AA</i><sup>∗</sup> = <i>A</i><sup>∗</sup><i>A</i></td> <td>They are the matrices to which the <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> applies. </td></tr> <tr> <td><a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">Orthogonal matrix</a></td> <td>A matrix whose inverse is equal to its <a href="/wiki/Transpose" title="Transpose">transpose</a>, <i>A</i><sup>−1</sup> = <i>A</i><sup><i>T</i></sup>.</td> <td>They form the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>. </td></tr> <tr> <td><a href="/wiki/Orthonormal_matrix" class="mw-redirect" title="Orthonormal matrix">Orthonormal matrix</a></td> <td>A matrix whose columns are <a href="/wiki/Orthonormal" class="mw-redirect" title="Orthonormal">orthonormal</a> vectors.</td> <td> </td></tr> <tr> <td><a href="/wiki/Partial_isometry" title="Partial isometry">Partially Isometric matrix</a></td> <td>A matrix that is an <a href="/wiki/Isometry" title="Isometry">isometry</a> on the <a href="/wiki/Orthogonal_complement" title="Orthogonal complement">orthogonal complement</a> of its <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a>. Equivalently, a matrix that satisfies <i>AA</i><sup>*</sup><i>A</i> = <i>A</i>.</td> <td>Equivalently, a matrix with <a href="/wiki/Singular_value" title="Singular value">singular values</a> that are either 0 or 1. </td></tr> <tr> <td><a href="/wiki/Singular_matrix" class="mw-redirect" title="Singular matrix">Singular matrix</a></td> <td>A square matrix that is not invertible.</td> <td> </td></tr> <tr> <td><a href="/wiki/Unimodular_matrix" title="Unimodular matrix">Unimodular matrix</a></td> <td>An invertible matrix with entries in the integers (<a href="/wiki/Integer_matrix" title="Integer matrix">integer matrix</a>)</td> <td>Necessarily the determinant is +1 or −1. </td></tr> <tr> <td><a href="/wiki/Unipotent_matrix" class="mw-redirect" title="Unipotent matrix">Unipotent matrix</a></td> <td>A square matrix with all eigenvalues equal to 1.</td> <td>Equivalently, <span class="nowrap"><i>A</i> − <i>I</i></span> is nilpotent. See also <a href="/wiki/Unipotent_group" class="mw-redirect" title="Unipotent group">unipotent group</a>. </td></tr> <tr> <td><a href="/wiki/Unitary_matrix" title="Unitary matrix">Unitary matrix</a> </td> <td>A square matrix whose inverse is equal to its <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>, <i>A</i><sup>−1</sup> = <i>A</i><sup>*</sup>. </td> <td> </td></tr> <tr> <td><a href="/wiki/Totally_unimodular_matrix" class="mw-redirect" title="Totally unimodular matrix">Totally unimodular matrix</a></td> <td>A matrix for which every non-singular square submatrix is <a href="/wiki/Unimodular_matrix" title="Unimodular matrix">unimodular</a>. This has some implications in the <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a> <a href="/wiki/Linear_programming_relaxation" title="Linear programming relaxation">relaxation</a> of an <a href="/wiki/Integer_program" class="mw-redirect" title="Integer program">integer program</a>.</td> <td> </td></tr> <tr> <td><a href="/wiki/Weighing_matrix" title="Weighing matrix">Weighing matrix</a></td> <td>A square matrix the entries of which are in <span class="nowrap">{0, 1, −1}</span>, such that <i>AA</i><sup>T</sup> = <i>wI</i> for some positive integer <i>w</i>.</td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Matrices_with_conditions_on_eigenvalues_or_eigenvectors">Matrices with conditions on eigenvalues or eigenvectors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=4" title="Edit section: Matrices with conditions on eigenvalues or eigenvectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable"> <tbody><tr> <th>Name</th> <th>Explanation</th> <th>Notes </th></tr> <tr> <td><a href="/wiki/Convergent_matrix" title="Convergent matrix">Convergent matrix</a></td> <td>A square matrix whose successive powers approach the <a href="/wiki/Zero_matrix" title="Zero matrix">zero matrix</a>.</td> <td>Its <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> have magnitude less than one. </td></tr> <tr> <td><a href="/wiki/Defective_matrix" title="Defective matrix">Defective matrix</a></td> <td>A square matrix that does not have a complete basis of <a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvectors</a>, and is thus not <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a>.</td> <td> </td></tr> <tr> <td><a href="/w/index.php?title=Derogatory_matrix&action=edit&redlink=1" class="new" title="Derogatory matrix (page does not exist)">Derogatory matrix</a> </td> <td>A square matrix whose <a href="/wiki/Minimal_polynomial_(linear_algebra)" title="Minimal polynomial (linear algebra)">minimal polynomial</a> is of order less than <i>n</i>. Equivalently, at least one of its eigenvalues has at least two <a href="/wiki/Jordan_block" class="mw-redirect" title="Jordan block">Jordan blocks</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </td> <td> </td></tr> <tr> <td><a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">Diagonalizable matrix</a></td> <td>A square matrix <a href="/wiki/Similar_matrix" class="mw-redirect" title="Similar matrix">similar</a> to a diagonal matrix.</td> <td>It has an <a href="/wiki/Eigenbasis" class="mw-redirect" title="Eigenbasis">eigenbasis</a>, that is, a complete set of <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> eigenvectors. </td></tr> <tr> <td><a href="/wiki/Hurwitz-stable_matrix" title="Hurwitz-stable matrix">Hurwitz matrix</a></td> <td>A matrix whose eigenvalues have strictly negative real part. A stable system of differential equations may be represented by a Hurwitz matrix.</td> <td> </td></tr> <tr> <td><a href="/wiki/M-matrix" title="M-matrix">M-matrix</a></td> <td>A Z-matrix with eigenvalues whose real parts are nonnegative.</td> <td> </td></tr> <tr> <td><a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">Positive-definite matrix</a></td> <td>A Hermitian matrix with every eigenvalue positive.</td> <td> </td></tr> <tr> <td><a href="/wiki/Stability_matrix" class="mw-redirect" title="Stability matrix">Stability matrix</a></td> <td></td> <td>Synonym for <a href="/wiki/Hurwitz-stable_matrix" title="Hurwitz-stable matrix">Hurwitz matrix</a>. </td></tr> <tr> <td><a href="/wiki/Stieltjes_matrix" title="Stieltjes matrix">Stieltjes matrix</a></td> <td>A real symmetric positive definite matrix with nonpositive off-diagonal entries.</td> <td>Special case of an <a href="/wiki/M-matrix" title="M-matrix">M-matrix</a>. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Matrices_generated_by_specific_data">Matrices generated by specific data</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=5" title="Edit section: Matrices generated by specific data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable"> <tbody><tr> <th>Name</th> <th>Definition</th> <th>Comments </th></tr> <tr> <td><a href="/wiki/Adjugate_matrix" title="Adjugate matrix">Adjugate matrix</a></td> <td><a href="/wiki/Transpose" title="Transpose">Transpose</a> of the <a href="/wiki/Cofactor_matrix" class="mw-redirect" title="Cofactor matrix">cofactor matrix</a></td> <td>The <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse of a matrix</a> is its adjugate matrix divided by its <a href="/wiki/Determinant" title="Determinant">determinant</a> </td></tr> <tr> <td><a href="/wiki/Augmented_matrix" title="Augmented matrix">Augmented matrix</a></td> <td>Matrix whose rows are concatenations of the rows of two smaller matrices</td> <td>Used for performing the same <a href="/wiki/Row_operations" class="mw-redirect" title="Row operations">row operations</a> on two matrices </td></tr> <tr> <td><a href="/wiki/B%C3%A9zout_matrix" title="Bézout matrix">Bézout matrix</a></td> <td>Square matrix whose <a href="/wiki/Determinant" title="Determinant">determinant</a> is the <a href="/wiki/Resultant" title="Resultant">resultant</a> of two polynomials</td> <td>See also <a href="/wiki/Sylvester_matrix" title="Sylvester matrix">Sylvester matrix</a> </td></tr> <tr> <td><a href="/wiki/Carleman_matrix" title="Carleman matrix">Carleman matrix</a></td> <td>Infinite matrix of the <a href="/wiki/Taylor_coefficient" class="mw-redirect" title="Taylor coefficient">Taylor coefficients</a> of an <a href="/wiki/Analytic_function" title="Analytic function">analytic function</a> and its integer powers</td> <td>The composition of two functions can be expressed as the product of their Carleman matrices </td></tr> <tr> <td><a href="/wiki/Cartan_matrix" title="Cartan matrix">Cartan matrix</a></td> <td>A matrix associated with either a finite-dimensional <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a>, or a <a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">semisimple Lie algebra</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Cofactor_matrix" class="mw-redirect" title="Cofactor matrix">Cofactor matrix</a></td> <td>Formed by the <a href="/wiki/Cofactor_(linear_algebra)" class="mw-redirect" title="Cofactor (linear algebra)">cofactors</a> of a square matrix, that is, the signed <a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">minors</a>, of the matrix</td> <td><a href="/wiki/Transpose" title="Transpose">Transpose</a> of the <a href="/wiki/Adjugate_matrix" title="Adjugate matrix">Adjugate matrix</a> </td></tr> <tr> <td><a href="/wiki/Companion_matrix" title="Companion matrix">Companion matrix</a></td> <td>A matrix having the coefficients of a polynomial as last column, and having the polynomial as its <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Coxeter_matrix" class="mw-redirect" title="Coxeter matrix">Coxeter matrix</a></td> <td>A matrix which describes the relations between the <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involutions</a> that generate a <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter group</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Distance_matrix" title="Distance matrix">Distance matrix</a></td> <td>The square matrix formed by the pairwise distances of a set of <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a></td> <td><a href="/wiki/Euclidean_distance_matrix" title="Euclidean distance matrix">Euclidean distance matrix</a> is a special case </td></tr> <tr> <td><a href="/wiki/Euclidean_distance_matrix" title="Euclidean distance matrix">Euclidean distance matrix</a></td> <td>A matrix that describes the pairwise distances between <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a></td> <td>See also <a href="/wiki/Distance_matrix" title="Distance matrix">distance matrix</a> </td></tr> <tr> <td><a href="/wiki/Fundamental_matrix_(linear_differential_equation)" title="Fundamental matrix (linear differential equation)">Fundamental matrix</a></td> <td>The matrix formed from the fundamental solutions of a <a href="/wiki/System_of_linear_differential_equations" class="mw-redirect" title="System of linear differential equations">system of linear differential equations</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Generator_matrix" title="Generator matrix">Generator matrix</a></td> <td>In <a href="/wiki/Coding_theory" title="Coding theory">Coding theory</a>, a matrix whose rows <a href="/wiki/Linear_span" title="Linear span">span</a> a <a href="/wiki/Linear_code" title="Linear code">linear code</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Gramian_matrix" class="mw-redirect" title="Gramian matrix">Gramian matrix</a></td> <td>The symmetric matrix of the pairwise <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner products</a> of a set of vectors in an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></td> <td>The square matrix of <a href="/wiki/Partial_derivative" title="Partial derivative">second partial derivatives</a> of a <a href="/wiki/Function_of_several_variables" class="mw-redirect" title="Function of several variables">function of several variables</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Householder_transformation#Householder_matrix" title="Householder transformation">Householder matrix</a></td> <td>The matrix of a <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> with respect to a <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> passing through the origin</td> <td> </td></tr> <tr> <td><a href="/wiki/Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a></td> <td>The matrix of the partial derivatives of a <a href="/wiki/Function_of_several_variables" class="mw-redirect" title="Function of several variables">function of several variables</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Moment_matrix" title="Moment matrix">Moment matrix</a> </td> <td></td> <td>Used in <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Sum-of-squares_optimization" title="Sum-of-squares optimization">Sum-of-squares optimization</a> </td></tr> <tr> <td><a href="/wiki/Payoff_matrix" class="mw-redirect" title="Payoff matrix">Payoff matrix</a></td> <td>A matrix in <a href="/wiki/Game_theory" title="Game theory">game theory</a> and <a href="/wiki/Economics" title="Economics">economics</a>, that represents the payoffs in a <a href="/wiki/Normal_form_game" class="mw-redirect" title="Normal form game">normal form game</a> where players move simultaneously</td> <td> </td></tr> <tr> <td><a href="/wiki/Pick_matrix" class="mw-redirect" title="Pick matrix">Pick matrix</a></td> <td>A matrix that occurs in the study of analytical interpolation problems</td> <td> </td></tr> <tr> <td><a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation matrix</a></td> <td>A matrix representing a <a href="/wiki/Rotation_(geometry)" class="mw-redirect" title="Rotation (geometry)">rotation</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Seifert_matrix" class="mw-redirect" title="Seifert matrix">Seifert matrix</a></td> <td>A matrix in <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>, primarily for the algebraic analysis of topological properties of knots and links.</td> <td><a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a> </td></tr> <tr> <td><a href="/wiki/Shear_matrix" class="mw-redirect" title="Shear matrix">Shear matrix</a></td> <td>The matrix of a <a href="/wiki/Shear_transformation" class="mw-redirect" title="Shear transformation">shear transformation</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Similarity_matrix" class="mw-redirect" title="Similarity matrix">Similarity matrix</a></td> <td>A matrix of scores which express the similarity between two data points</td> <td><a href="/wiki/Sequence_alignment" title="Sequence alignment">Sequence alignment</a> </td></tr> <tr> <td><a href="/wiki/Sylvester_matrix" title="Sylvester matrix">Sylvester matrix</a></td> <td>A square matrix whose entries come from the coefficients of two <a href="/wiki/Polynomials" class="mw-redirect" title="Polynomials">polynomials</a></td> <td>The Sylvester matrix is nonsingular if and only if the two polynomials are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> to each other </td></tr> <tr> <td><a href="/wiki/Symplectic_matrix" title="Symplectic matrix">Symplectic matrix</a></td> <td>The real matrix of a <a href="/wiki/Symplectic_transformation" class="mw-redirect" title="Symplectic transformation">symplectic transformation</a></td> <td> </td></tr> <tr> <td><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation matrix</a></td> <td>The matrix of a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> or a <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformation</a></td> <td> </td></tr> <tr> <td><a href="/w/index.php?title=Wedderburn_matrix&action=edit&redlink=1" class="new" title="Wedderburn matrix (page does not exist)">Wedderburn matrix</a></td> <td>A matrix of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-(y^{T}Ax)^{-1}Axy^{T}A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>A</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-(y^{T}Ax)^{-1}Axy^{T}A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5af3e89b1a1f8c5f4b4357e8d6cf5ba7c4f879f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.714ex; height:3.176ex;" alt="{\displaystyle A-(y^{T}Ax)^{-1}Axy^{T}A}"></span>, used for rank-reduction & biconjugate decompositions</td> <td>Analysis of matrix decompositions </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Matrices_used_in_statistics">Matrices used in statistics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=6" title="Edit section: Matrices used in statistics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following matrices find their main application in <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>. </p> <ul><li><a href="/w/index.php?title=Bernoulli_matrix&action=edit&redlink=1" class="new" title="Bernoulli matrix (page does not exist)">Bernoulli matrix</a> — a square matrix with entries +1, −1, with equal <a href="/wiki/Probability" title="Probability">probability</a> of each.</li> <li><a href="/wiki/Centering_matrix" title="Centering matrix">Centering matrix</a> — a matrix which, when multiplied with a vector, has the same effect as subtracting the mean of the components of the vector from every component.</li> <li><a href="/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">Correlation matrix</a> — a symmetric <i>n×n</i> matrix, formed by the pairwise <a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">correlation coefficients</a> of several <a href="/wiki/Random_variable" title="Random variable">random variables</a>.</li> <li><a href="/wiki/Covariance_matrix" title="Covariance matrix">Covariance matrix</a> — a symmetric <i>n×n</i> matrix, formed by the pairwise <a href="/wiki/Covariance" title="Covariance">covariances</a> of several random variables. Sometimes called a <i>dispersion matrix</i>.</li> <li><a href="/wiki/Dispersion_matrix" class="mw-redirect" title="Dispersion matrix">Dispersion matrix</a> — another name for a <i>covariance matrix</i>.</li> <li><a href="/wiki/Doubly_stochastic_matrix" title="Doubly stochastic matrix">Doubly stochastic matrix</a> — a non-negative matrix such that each row and each column sums to 1 (thus the matrix is both <i>left stochastic</i> and <i>right stochastic</i>)</li> <li><a href="/wiki/Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information matrix</a> — a matrix representing the variance of the partial derivative, with respect to a parameter, of the log of the likelihood function of a random variable.</li> <li><a href="/wiki/Hat_matrix" class="mw-redirect" title="Hat matrix">Hat matrix</a> — a square matrix used in statistics to relate fitted values to observed values.</li> <li><a href="/wiki/Orthostochastic_matrix" title="Orthostochastic matrix">Orthostochastic matrix</a> — doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix</li> <li><a href="/wiki/Precision_matrix" class="mw-redirect" title="Precision matrix">Precision matrix</a> — a symmetric <i>n×n</i> matrix, formed by inverting the <i>covariance matrix</i>. Also called the <i>information matrix</i>.</li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Stochastic matrix</a> — a <a href="/wiki/Non-negative" class="mw-redirect" title="Non-negative">non-negative</a> matrix describing a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a>. The sum of entries of any row is one.</li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Transition matrix</a> — a matrix representing the <a href="/wiki/Probabilities" class="mw-redirect" title="Probabilities">probabilities</a> of conditions changing from one state to another in a <a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Unistochastic_matrix" title="Unistochastic matrix">Unistochastic matrix</a> — a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix</li></ul> <div class="mw-heading mw-heading2"><h2 id="Matrices_used_in_graph_theory">Matrices used in graph theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=7" title="Edit section: Matrices used in graph theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following matrices find their main application in <a href="/wiki/Graph_theory" title="Graph theory">graph</a> and <a href="/wiki/Network_theory" title="Network theory">network theory</a>. </p> <ul><li><a href="/wiki/Adjacency_matrix" title="Adjacency matrix">Adjacency matrix</a> — a square matrix representing a graph, with <i>a<sub>ij</sub></i> non-zero if vertex <i>i</i> and vertex <i>j</i> are adjacent.</li> <li><a href="/wiki/Biadjacency_matrix" class="mw-redirect" title="Biadjacency matrix">Biadjacency matrix</a> — a special class of <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> that describes adjacency in <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite graphs</a>.</li> <li><a href="/wiki/Degree_matrix" title="Degree matrix">Degree matrix</a> — a diagonal matrix defining the degree of each <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertex</a> in a graph.</li> <li><a href="/wiki/Edmonds_matrix" title="Edmonds matrix">Edmonds matrix</a> — a square matrix of a bipartite graph.</li> <li><a href="/wiki/Incidence_matrix" title="Incidence matrix">Incidence matrix</a> — a matrix representing a relationship between two classes of objects (usually <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertices</a> and <a href="/wiki/Edge_(graph_theory)" class="mw-redirect" title="Edge (graph theory)">edges</a> in the context of graph theory).</li> <li><a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian matrix</a> — a matrix equal to the degree matrix minus the adjacency matrix for a graph, used to find the number of spanning trees in the graph.</li> <li><a href="/wiki/Seidel_adjacency_matrix" title="Seidel adjacency matrix">Seidel adjacency matrix</a> — a matrix similar to the usual <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> but with −1 for adjacency; +1 for nonadjacency; 0 on the diagonal.</li> <li><a href="/w/index.php?title=Skew-adjacency_matrix&action=edit&redlink=1" class="new" title="Skew-adjacency matrix (page does not exist)">Skew-adjacency matrix</a> — an <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> in which each non-zero <i>a<sub>ij</sub></i> is 1 or −1, accordingly as the direction <i>i → j</i> matches or opposes that of an initially specified orientation.</li> <li><a href="/wiki/Tutte_matrix" title="Tutte matrix">Tutte matrix</a> — a generalization of the Edmonds matrix for a balanced bipartite graph.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Matrices_used_in_science_and_engineering">Matrices used in science and engineering</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=8" title="Edit section: Matrices used in science and engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa matrix</a> — a unitary matrix used in <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a> to describe the strength of <i>flavour-changing</i> weak decays.</li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a> — a matrix describing the statistical state of a quantum system. <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a>, <a href="/wiki/Non-negative_matrix" class="mw-redirect" title="Non-negative matrix">non-negative</a> and with <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> 1.</li> <li><a href="/wiki/Fundamental_matrix_(computer_vision)" title="Fundamental matrix (computer vision)">Fundamental matrix (computer vision)</a> — a 3 × 3 matrix in <a href="/wiki/Computer_vision" title="Computer vision">computer vision</a> that relates corresponding points in stereo images.</li> <li><a href="/wiki/Fuzzy_associative_matrix" title="Fuzzy associative matrix">Fuzzy associative matrix</a> — a matrix in <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, used in machine learning processes.</li> <li><a href="/wiki/Gamma_matrices" title="Gamma matrices">Gamma matrices</a> — 4 × 4 matrices in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>.</li> <li><a href="/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann matrices</a> — a <a href="/wiki/Generalizations_of_Pauli_matrices" title="Generalizations of Pauli matrices">generalization of the Pauli matrices</a>; these matrices are one notable representation of the <a href="/wiki/Lie_group#The_Lie_algebra_associated_to_a_Lie_group" title="Lie group">infinitesimal generators</a> of the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a> SU(3).</li> <li><a href="/wiki/Hamiltonian_matrix" title="Hamiltonian matrix">Hamiltonian matrix</a> — a matrix used in a variety of fields, including <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and <a href="/wiki/Linear-quadratic_regulator" class="mw-redirect" title="Linear-quadratic regulator">linear-quadratic regulator</a> (LQR) systems.</li> <li><a href="/wiki/Irregular_matrix" title="Irregular matrix">Irregular matrix</a> — a matrix used in <a href="/wiki/Computer_science" title="Computer science">computer science</a> which has a varying number of elements in each row.</li> <li><a href="/wiki/Overlap_matrix" class="mw-redirect" title="Overlap matrix">Overlap matrix</a> — a type of <a href="/wiki/Gramian_matrix" class="mw-redirect" title="Gramian matrix">Gramian matrix</a>, used in <a href="/wiki/Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a> to describe the inter-relationship of a set of <a href="/wiki/Basis_vector" class="mw-redirect" title="Basis vector">basis vectors</a> of a <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum</a> system.</li> <li><a href="/wiki/S_matrix" class="mw-redirect" title="S matrix">S matrix</a> — a matrix in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> that connects asymptotic (infinite past and future) particle states.</li> <li><a href="/wiki/Scattering_matrix" class="mw-redirect" title="Scattering matrix">Scattering matrix</a> - a matrix in Microwave Engineering that describes how the power move in a multiport system.</li> <li><a href="/wiki/State-transition_matrix" title="State-transition matrix">State transition matrix</a> — exponent of state matrix in control systems.</li> <li><a href="/wiki/Substitution_matrix" title="Substitution matrix">Substitution matrix</a> — a matrix from <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>, which describes mutation rates of <a href="/wiki/Amino_acid" title="Amino acid">amino acid</a> or <a href="/wiki/DNA" title="DNA">DNA</a> sequences.</li> <li><a href="/wiki/Supnick_matrix" title="Supnick matrix">Supnick matrix</a> — a square matrix used in <a href="/wiki/Computer_science" title="Computer science">computer science</a>.</li> <li><a href="/wiki/Z-matrix_(chemistry)" title="Z-matrix (chemistry)">Z-matrix</a> — a matrix in <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, representing a molecule in terms of its relative atomic geometry.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Specific_matrices">Specific matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=9" title="Edit section: Specific matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Wilson_matrix" title="Wilson matrix">Wilson matrix</a>, a matrix used as an example for test purposes.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Other_matrix-related_terms_and_definitions">Other matrix-related terms and definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=10" title="Edit section: Other matrix-related terms and definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Jordan_canonical_form" class="mw-redirect" title="Jordan canonical form">Jordan canonical form</a> — an 'almost' diagonalised matrix, where the only non-zero elements appear on the lead and superdiagonals.</li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a> — two or more <a href="/wiki/Coordinate_vector" title="Coordinate vector">vectors</a> are linearly independent if there is no way to construct one from <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of the others.</li> <li><a href="/wiki/Matrix_exponential" title="Matrix exponential">Matrix exponential</a> — defined by the <a href="/wiki/Exponential_function#Formal_definition" title="Exponential function">exponential series</a>.</li> <li><a href="/wiki/Matrix_representation_of_conic_sections" title="Matrix representation of conic sections">Matrix representation of conic sections</a></li> <li><a href="/wiki/Pseudoinverse" class="mw-redirect" title="Pseudoinverse">Pseudoinverse</a> — a generalization of the <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse matrix</a>.</li> <li><a href="/wiki/Row_echelon_form" title="Row echelon form">Row echelon form</a> — a matrix in this form is the result of applying the <i>forward elimination</i> procedure to a matrix (as used in <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>).</li> <li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a> — the determinant of a matrix of functions and their derivatives such that row <i>n</i> is the (<i>n</i>−1)<sup>th</sup> derivative of row one.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Perfect_matrix" title="Perfect matrix">Perfect matrix</a></li></ul> <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 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class="reference-text">Hogben <a href="#CITEREFHogben2006">2006</a>, Ch. 31.3.</span> </li> <li id="cite_note-:1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output 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.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/MatrixMultiplication.html">"Matrix Multiplication"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-07</span></span>.</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+Multiplication&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMatrixMultiplication.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+named+matrices" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Non-derogatory_matrix">"Non-derogatory matrix - Encyclopedia of Mathematics"</a>. <i>encyclopediaofmath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=encyclopediaofmath.org&rft.atitle=Non-derogatory+matrix+-+Encyclopedia+of+Mathematics&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FNon-derogatory_matrix&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+named+matrices" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=List_of_named_matrices&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHogben2006" class="citation cs2"><a href="/wiki/Leslie_Hogben" title="Leslie Hogben">Hogben, Leslie</a> (2006), <i>Handbook of Linear Algebra (Discrete Mathematics and Its Applications)</i>, Boca Raton: Chapman & Hall/CRC, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-510-8" title="Special:BookSources/978-1-58488-510-8"><bdi>978-1-58488-510-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Linear+Algebra+%28Discrete+Mathematics+and+Its+Applications%29&rft.place=Boca+Raton&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2006&rft.isbn=978-1-58488-510-8&rft.aulast=Hogben&rft.aufirst=Leslie&rfr_id=info%3Asid%2Fen.wikipedia.org%3AList+of+named+matrices" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=List_of_named_matrices&oldid=1255572705">https://en.wikipedia.org/w/index.php?title=List_of_named_matrices&oldid=1255572705</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Mathematics-related_lists" title="Category:Mathematics-related lists">Mathematics-related lists</a></li><li><a href="/wiki/Category:Matrices" title="Category:Matrices">Matrices</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 5 November 2024, at 16:53<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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