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class="vector-toc-numb">2</span> <span>Decompositions related to solving systems of linear equations</span> </div> </a> <button aria-controls="toc-Decompositions_related_to_solving_systems_of_linear_equations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Decompositions related to solving systems of linear equations subsection</span> </button> <ul id="toc-Decompositions_related_to_solving_systems_of_linear_equations-sublist" class="vector-toc-list"> <li id="toc-LU_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#LU_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>LU decomposition</span> </div> </a> <ul id="toc-LU_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-LU_reduction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#LU_reduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>LU reduction</span> </div> </a> <ul id="toc-LU_reduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Block_LU_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Block_LU_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Block LU decomposition</span> </div> </a> <ul id="toc-Block_LU_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rank_factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rank_factorization"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Rank factorization</span> </div> </a> <ul id="toc-Rank_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cholesky_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cholesky_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Cholesky decomposition</span> </div> </a> <ul id="toc-Cholesky_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-QR_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#QR_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>QR decomposition</span> </div> </a> <ul id="toc-QR_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-RRQR_factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#RRQR_factorization"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>RRQR factorization</span> </div> </a> <ul id="toc-RRQR_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpolative_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpolative_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Interpolative decomposition</span> </div> </a> <ul id="toc-Interpolative_decomposition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Decompositions_based_on_eigenvalues_and_related_concepts" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Decompositions_based_on_eigenvalues_and_related_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Decompositions based on eigenvalues and related concepts</span> </div> </a> <button aria-controls="toc-Decompositions_based_on_eigenvalues_and_related_concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Decompositions based on eigenvalues and related concepts subsection</span> </button> <ul id="toc-Decompositions_based_on_eigenvalues_and_related_concepts-sublist" class="vector-toc-list"> <li id="toc-Eigendecomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigendecomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Eigendecomposition</span> </div> </a> <ul id="toc-Eigendecomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jordan_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jordan_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Jordan decomposition</span> </div> </a> <ul id="toc-Jordan_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Schur_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schur_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Schur decomposition</span> </div> </a> <ul id="toc-Schur_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_Schur_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_Schur_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Real Schur decomposition</span> </div> </a> <ul id="toc-Real_Schur_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-QZ_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#QZ_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>QZ decomposition</span> </div> </a> <ul id="toc-QZ_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Takagi's_factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Takagi's_factorization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Takagi's factorization</span> </div> </a> <ul id="toc-Takagi's_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Singular_value_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Singular_value_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Singular value decomposition</span> </div> </a> <ul id="toc-Singular_value_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scale-invariant_decompositions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scale-invariant_decompositions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Scale-invariant decompositions</span> </div> </a> <ul id="toc-Scale-invariant_decompositions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hessenberg_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hessenberg_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Hessenberg decomposition</span> </div> </a> <ul id="toc-Hessenberg_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complete_orthogonal_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complete_orthogonal_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.10</span> <span>Complete orthogonal decomposition</span> </div> </a> <ul id="toc-Complete_orthogonal_decomposition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_decompositions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_decompositions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other decompositions</span> </div> </a> <button aria-controls="toc-Other_decompositions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Other decompositions subsection</span> </button> <ul id="toc-Other_decompositions-sublist" class="vector-toc-list"> <li id="toc-Polar_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polar_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Polar decomposition</span> </div> </a> <ul id="toc-Polar_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_polar_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_polar_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Algebraic polar decomposition</span> </div> </a> <ul id="toc-Algebraic_polar_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mostow's_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mostow's_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Mostow's decomposition</span> </div> </a> <ul id="toc-Mostow's_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sinkhorn_normal_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sinkhorn_normal_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Sinkhorn normal form</span> </div> </a> <ul id="toc-Sinkhorn_normal_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sectoral_decomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sectoral_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Sectoral decomposition</span> </div> </a> <ul id="toc-Sectoral_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Williamson's_normal_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Williamson's_normal_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Williamson's normal form</span> </div> </a> <ul id="toc-Williamson's_normal_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_square_root" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_square_root"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Matrix square root</span> </div> </a> <ul id="toc-Matrix_square_root-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> 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<div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Matrix decomposition</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 19 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-19" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%81%D9%83%D9%8A%D9%83_%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9" title="تفكيك مصفوفة – Arabic" lang="ar" hreflang="ar" data-title="تفكيك مصفوفة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Descomposici%C3%B3_de_matrius" title="Descomposició de matrius – Catalan" lang="ca" hreflang="ca" data-title="Descomposició de matrius" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_%D1%81%D0%B0%D1%80%C4%83%D0%BC%C4%95" title="Матрица сарăмĕ – Chuvash" lang="cv" hreflang="cv" data-title="Матрица сарăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Factorizaci%C3%B3n_de_matrices" title="Factorización de matrices – Spanish" lang="es" hreflang="es" data-title="Factorización de matrices" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%AC%D8%B2%DB%8C%D9%87_%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3" title="تجزیه ماتریس – Persian" lang="fa" hreflang="fa" data-title="تجزیه ماتریس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9composition_d%27une_matrice" title="Décomposition d'une matrice – French" lang="fr" hreflang="fr" data-title="Décomposition d'une matrice" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Descomposici%C3%B3n_de_matrices" title="Descomposición de matrices – Galician" lang="gl" hreflang="gl" data-title="Descomposición de matrices" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%96%89%EB%A0%AC_%EB%B6%84%ED%95%B4" title="행렬 분해 – Korean" lang="ko" hreflang="ko" data-title="행렬 분해" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fylkjali%C3%B0un" title="Fylkjaliðun – Icelandic" lang="is" hreflang="is" data-title="Fylkjaliðun" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Decomposizione_di_una_matrice" title="Decomposizione di una matrice – Italian" lang="it" hreflang="it" data-title="Decomposizione di una matrice" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Matrixdecompositie" title="Matrixdecompositie – Dutch" lang="nl" hreflang="nl" data-title="Matrixdecompositie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A1%8C%E5%88%97%E3%81%AE%E5%88%86%E8%A7%A3" title="行列の分解 – Japanese" lang="ja" hreflang="ja" data-title="行列の分解" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_macierzy" title="Rozkład macierzy – Polish" lang="pl" hreflang="pl" data-title="Rozkład macierzy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Decomposi%C3%A7%C3%A3o_matricial" title="Decomposição matricial – Portuguese" lang="pt" hreflang="pt" data-title="Decomposição matricial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D0%BB%D0%BE%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8B" title="Разложение матрицы – Russian" lang="ru" hreflang="ru" data-title="Разложение матрицы" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matrisfaktorisering" title="Matrisfaktorisering – Swedish" lang="sv" hreflang="sv" data-title="Matrisfaktorisering" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%BA%D0%BB%D0%B0%D0%B4_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%96" title="Розклад матриці – Ukrainian" lang="uk" hreflang="uk" data-title="Розклад матриці" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_r%C3%A3_ma_tr%E1%BA%ADn" title="Phân rã ma trận – Vietnamese" lang="vi" hreflang="vi" data-title="Phân rã ma trận" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3" title="矩阵分解 – Chinese" lang="zh" hreflang="zh" data-title="矩阵分解" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1361088#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Matrix_factorization_of_a_polynomial" title="Matrix factorization of a polynomial">matrix factorization of a polynomial</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Representation of a matrix as a product</div> <p>In the <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> discipline of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, a <b>matrix decomposition</b> or <b>matrix factorization</b> is a <a href="/wiki/Factorization" title="Factorization">factorization</a> of a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=1" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>, different decompositions are used to implement efficient matrix <a href="/wiki/Algorithm" title="Algorithm">algorithms</a>. </p><p>For instance, when solving a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a> <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\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <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 A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span>, the matrix <i>A</i> can be decomposed via the <a href="/wiki/LU_decomposition" title="LU decomposition">LU decomposition</a>. The LU decomposition factorizes a matrix into a <a href="/wiki/Lower_triangular_matrix" class="mw-redirect" title="Lower triangular matrix">lower triangular matrix</a> <i>L</i> and an <a href="/wiki/Upper_triangular_matrix" class="mw-redirect" title="Upper triangular matrix">upper triangular matrix</a> <i>U</i>. The systems <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 L(U\mathbf {x} )=\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(U\mathbf {x} )=\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55c9cfdfe772f24e24bb4ba0307adc167681d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.169ex; height:2.843ex;" alt="{\displaystyle L(U\mathbf {x} )=\mathbf {b} }"></span> and <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 U\mathbf {x} =L^{-1}\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msup> <mi>L</mi> <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 U\mathbf {x} =L^{-1}\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/254570218cd039cd14aaf3bb030de8a21e8124c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.693ex; height:2.676ex;" alt="{\displaystyle U\mathbf {x} =L^{-1}\mathbf {b} }"></span> require fewer additions and multiplications to solve, compared with the original system <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\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <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 A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span>, though one might require significantly more digits in inexact arithmetic such as <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a>. </p><p>Similarly, the <a href="/wiki/QR_decomposition" title="QR decomposition">QR decomposition</a> expresses <i>A</i> as <i>QR</i> with <i>Q</i> an <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrix</a> and <i>R</i> an upper triangular matrix. The system <i>Q</i>(<i>R</i><b>x</b>) = <b>b</b> is solved by <i>R</i><b>x</b> = <i>Q</i><sup>T</sup><b>b</b> = <b>c</b>, and the system <i>R</i><b>x</b> = <b>c</b> is solved by '<a href="/wiki/Triangular_matrix#Forward_and_back_substitution" title="Triangular matrix">back substitution</a>'. The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is <a href="/wiki/Numerically_stable" class="mw-redirect" title="Numerically stable">numerically stable</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Decompositions_related_to_solving_systems_of_linear_equations">Decompositions related to solving systems of linear equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=2" title="Edit section: Decompositions related to solving systems of linear equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="LU_decomposition">LU decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=3" title="Edit section: LU decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/LU_decomposition" title="LU decomposition">LU decomposition</a></div> <ul><li>Traditionally applicable to: <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> <i>A</i>, although rectangular matrices can be applicable.<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup></li> <li>Decomposition: <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=LU}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>L</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=LU}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9af953d6ffa260c3f7ffa75edc63edc006c64d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.207ex; height:2.176ex;" alt="{\displaystyle A=LU}"></span>, where <i>L</i> is <a href="/wiki/Triangular_matrix" title="Triangular matrix">lower triangular</a> and <i>U</i> is <a href="/wiki/Triangular_matrix" title="Triangular matrix">upper triangular</a>.</li> <li>Related: the <a href="/wiki/LDU_decomposition" class="mw-redirect" title="LDU decomposition"><i>LDU</i> decomposition</a> is <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=LDU}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>L</mi> <mi>D</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=LDU}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9854740b03dca7ae7a902656f1056d505ca6c694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.131ex; height:2.176ex;" alt="{\displaystyle A=LDU}"></span>, where <i>L</i> is <a href="/wiki/Triangular_matrix" title="Triangular matrix">lower triangular</a> with ones on the diagonal, <i>U</i> is <a href="/wiki/Triangular_matrix" title="Triangular matrix">upper triangular</a> with ones on the diagonal, and <i>D</i> is a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a>.</li> <li>Related: the <a href="/wiki/LUP_decomposition" class="mw-redirect" title="LUP decomposition"><i>LUP</i> decomposition</a> is <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 PA=LU}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>A</mi> <mo>=</mo> <mi>L</mi> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PA=LU}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/401e8d7252876c4c8195094348fde42dbf8a6dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.952ex; height:2.176ex;" alt="{\displaystyle PA=LU}"></span>, where <i>L</i> is <a href="/wiki/Triangular_matrix" title="Triangular matrix">lower triangular</a>, <i>U</i> is <a href="/wiki/Triangular_matrix" title="Triangular matrix">upper triangular</a>, and <i>P</i> is a <a href="/wiki/Permutation_matrix" title="Permutation matrix">permutation matrix</a>.</li> <li>Existence: An LUP decomposition exists for any square matrix <i>A</i>. When <i>P</i> is an <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>, the LUP decomposition reduces to the LU decomposition.</li> <li>Comments: The LUP and LU decompositions are useful in solving an <i>n</i>-by-<i>n</i> system of linear equations <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\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <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 A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span>. These decompositions summarize the process of <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> in matrix form. Matrix <i>P</i> represents any row interchanges carried out in the process of Gaussian elimination. If Gaussian elimination produces the <a href="/wiki/Row_echelon_form" title="Row echelon form">row echelon form</a> without requiring any row interchanges, then <i>P</i> = <i>I</i>, so an LU decomposition exists.</li></ul> <div class="mw-heading mw-heading3"><h3 id="LU_reduction">LU reduction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=4" title="Edit section: LU reduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/LU_reduction" title="LU reduction">LU reduction</a></div> <div class="mw-heading mw-heading3"><h3 id="Block_LU_decomposition">Block LU decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=5" title="Edit section: Block LU decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Block_LU_decomposition" title="Block LU decomposition">Block LU decomposition</a></div> <div class="mw-heading mw-heading3"><h3 id="Rank_factorization">Rank factorization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=6" title="Edit section: Rank factorization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Rank_factorization" title="Rank factorization">Rank factorization</a></div> <ul><li>Applicable to: <i>m</i>-by-<i>n</i> matrix <i>A</i> of rank <i>r</i></li> <li>Decomposition: <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=CF}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>C</mi> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=CF}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3210f2dcdfae81b74d6e0d8d58898a9f3ad51cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.349ex; height:2.176ex;" alt="{\displaystyle A=CF}"></span> where <i>C</i> is an <i>m</i>-by-<i>r</i> full column rank matrix and <i>F</i> is an <i>r</i>-by-<i>n</i> full row rank matrix</li> <li>Comment: The rank factorization can be used to <a href="/wiki/Moore%E2%80%93Penrose_pseudoinverse#Rank_decomposition" class="mw-redirect" title="Moore–Penrose pseudoinverse">compute the Moore–Penrose pseudoinverse</a> of <i>A</i>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> which one can apply to <a href="/wiki/Moore%E2%80%93Penrose_pseudoinverse#Obtaining_all_solutions_of_a_linear_system" class="mw-redirect" title="Moore–Penrose pseudoinverse">obtain all solutions of the linear system</a> <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\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <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 A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Cholesky_decomposition">Cholesky decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=7" title="Edit section: Cholesky decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Cholesky_decomposition" title="Cholesky decomposition">Cholesky decomposition</a></div> <ul><li>Applicable to: <a href="/wiki/Square_matrix" title="Square matrix">square</a>, <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">hermitian</a>, <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive definite</a> 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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span></li> <li>Decomposition: <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=U^{*}U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=U^{*}U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe78f8caa9baac5bd67a9c24d9c30e5d8935be5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.343ex;" alt="{\displaystyle A=U^{*}U}"></span>, 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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is upper triangular with real positive diagonal entries</li> <li>Comment: if the 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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is Hermitian and positive semi-definite, then it has a decomposition 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=U^{*}U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=U^{*}U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe78f8caa9baac5bd67a9c24d9c30e5d8935be5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.343ex;" alt="{\displaystyle A=U^{*}U}"></span> if the diagonal entries of <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> are allowed to be zero</li> <li>Uniqueness: for positive definite matrices Cholesky decomposition is unique. However, it is not unique in the positive semi-definite case.</li> <li>Comment: if <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is real and symmetric, <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> has all real elements</li> <li>Comment: An alternative is the <a href="/wiki/LDL_decomposition" class="mw-redirect" title="LDL decomposition">LDL decomposition</a>, which can avoid extracting square roots.</li></ul> <div class="mw-heading mw-heading3"><h3 id="QR_decomposition">QR decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=8" title="Edit section: QR decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/QR_decomposition" title="QR decomposition">QR decomposition</a></div> <ul><li>Applicable to: <i>m</i>-by-<i>n</i> matrix <i>A</i> with linearly independent columns</li> <li>Decomposition: <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=QR}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>Q</mi> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=QR}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d3eb7a53055abd0e9112408dbb2423035dfc72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.444ex; height:2.509ex;" alt="{\displaystyle A=QR}"></span> 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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a> of size <i>m</i>-by-<i>m</i>, and <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is an <a href="/wiki/Triangular_matrix" title="Triangular matrix">upper triangular</a> matrix of size <i>m</i>-by-<i>n</i></li> <li>Uniqueness: In general it is not unique, but if <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is of full <a href="/wiki/Matrix_rank" class="mw-redirect" title="Matrix rank">rank</a>, then there exists a single <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> that has all positive diagonal elements. If <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is square, also <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is unique.</li> <li>Comment: The QR decomposition provides an effective way to solve the system of equations <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\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <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 A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span>. The fact 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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a> means 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 Q^{\mathrm {T} }Q=I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> <mo>=</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{\mathrm {T} }Q=I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8c59b38159f56ce9ae0bf267460a83ca4b8710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.366ex; height:3.009ex;" alt="{\displaystyle Q^{\mathrm {T} }Q=I}"></span>, so 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 A\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <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 A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span> is equivalent to <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 R\mathbf {x} =Q^{\mathsf {T}}\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\mathbf {x} =Q^{\mathsf {T}}\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4689eb786ec3a1e3a46389cec76b5717c53c0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.949ex; height:3.009ex;" alt="{\displaystyle R\mathbf {x} =Q^{\mathsf {T}}\mathbf {b} }"></span>, which is very easy to solve since <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="RRQR_factorization">RRQR factorization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=9" title="Edit section: RRQR factorization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/RRQR_factorization" title="RRQR factorization">RRQR factorization</a></div> <div class="mw-heading mw-heading3"><h3 id="Interpolative_decomposition">Interpolative decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=10" title="Edit section: Interpolative decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Interpolative_decomposition" title="Interpolative decomposition">Interpolative decomposition</a></div> <div class="mw-heading mw-heading2"><h2 id="Decompositions_based_on_eigenvalues_and_related_concepts">Decompositions based on eigenvalues and related concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=11" title="Edit section: Decompositions based on eigenvalues and related concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Eigendecomposition">Eigendecomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=12" title="Edit section: Eigendecomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Eigendecomposition_(matrix)" class="mw-redirect" title="Eigendecomposition (matrix)">Eigendecomposition (matrix)</a></div> <ul><li>Also called <i><a href="/wiki/Spectral_decomposition_(Matrix)" class="mw-redirect" title="Spectral decomposition (Matrix)">spectral decomposition</a></i>.</li> <li>Applicable to: <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> <i>A</i> with linearly independent eigenvectors (not necessarily distinct eigenvalues).</li> <li>Decomposition: <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=VDV^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>V</mi> <mi>D</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=VDV^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6507fc0061185accb5c1a67516910e9a20a1848" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.803ex; height:2.676ex;" alt="{\displaystyle A=VDV^{-1}}"></span>, where <i>D</i> is a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> formed from the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of <i>A</i>, and the columns of <i>V</i> are the corresponding <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> of <i>A</i>.</li> <li>Existence: An <i>n</i>-by-<i>n</i> matrix <i>A</i> always has <i>n</i> (complex) eigenvalues, which can be ordered (in more than one way) to form an <i>n</i>-by-<i>n</i> diagonal matrix <i>D</i> and a corresponding matrix of nonzero columns <i>V</i> that satisfies the <a href="/wiki/Eigenvalue_equation" class="mw-redirect" title="Eigenvalue equation">eigenvalue equation</a> <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 AV=VD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>V</mi> <mo>=</mo> <mi>V</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AV=VD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3718928866c625f4f856124808ecc50530a80d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.34ex; height:2.176ex;" alt="{\displaystyle AV=VD}"></span>. <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is invertible if and only if the <i>n</i> eigenvectors are <a href="/wiki/Linear_independence" title="Linear independence">linearly independent</a> (that is, each eigenvalue has <a href="/wiki/Geometric_multiplicity" class="mw-redirect" title="Geometric multiplicity">geometric multiplicity</a> equal to its <a href="/wiki/Algebraic_multiplicity" class="mw-redirect" title="Algebraic multiplicity">algebraic multiplicity</a>). A sufficient (but not necessary) condition for this to happen is that all the eigenvalues are different (in this case geometric and algebraic multiplicity are equal to 1)</li> <li>Comment: One can always normalize the eigenvectors to have length one (see the definition of the eigenvalue equation)</li> <li>Comment: Every <a href="/wiki/Normal_matrix" title="Normal matrix">normal matrix</a> <i>A</i> (that is, matrix for which <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 AA^{*}=A^{*}A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA^{*}=A^{*}A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdd70c75482d4ec1548cc0a2095f5d9ad3970e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.179ex; height:2.343ex;" alt="{\displaystyle AA^{*}=A^{*}A}"></span>, 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 A^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="{\displaystyle A^{*}}"></span> is a <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>) can be eigendecomposed. For a <a href="/wiki/Normal_matrix" title="Normal matrix">normal matrix</a> <i>A</i> (and only for a normal matrix), the eigenvectors can also be made orthonormal (<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 VV^{*}=I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle VV^{*}=I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/273f98840b28a3b8a7179f3b0bd5a53559b68ddb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.029ex; height:2.343ex;" alt="{\displaystyle VV^{*}=I}"></span>) and the eigendecomposition reads as <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=VDV^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>V</mi> <mi>D</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=VDV^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c483f0f44b7c86d07b1af5db109c426d389d23b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.524ex; height:2.343ex;" alt="{\displaystyle A=VDV^{*}}"></span>. In particular all <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary</a>, <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a>, or <a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">skew-Hermitian</a> (in the real-valued case, all <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a>, <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>, or <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric</a>, respectively) matrices are normal and therefore possess this property.</li> <li>Comment: For any real <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a> <i>A</i>, the eigendecomposition always exists and can be written as <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=VDV^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>V</mi> <mi>D</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=VDV^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cce0dc345948e787d06dccbd8d5ff7b7b9631f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.822ex; height:2.676ex;" alt="{\displaystyle A=VDV^{\mathsf {T}}}"></span>, where both <i>D</i> and <i>V</i> are real-valued.</li> <li>Comment: The eigendecomposition is useful for understanding the solution of a system of linear ordinary differential equations or linear difference equations. For example, the difference equation <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_{t+1}=Ax_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>A</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{t+1}=Ax_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0c589cff900fe424eac0f22c9874bbb1f080b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.253ex; height:2.509ex;" alt="{\displaystyle x_{t+1}=Ax_{t}}"></span> starting from the initial condition <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_{0}=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded39e6ddcee68d613d246517365826562cf426f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.489ex; height:2.009ex;" alt="{\displaystyle x_{0}=c}"></span> is solved by <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_{t}=A^{t}c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{t}=A^{t}c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dcc062c181e4cf81bd437d6eb220df41ae8ea3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.83ex; height:2.843ex;" alt="{\displaystyle x_{t}=A^{t}c}"></span>, which is equivalent to <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_{t}=VD^{t}V^{-1}c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>V</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{t}=VD^{t}V^{-1}c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f7661651e900ca72d29a00708a6985855635e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.048ex; height:3.009ex;" alt="{\displaystyle x_{t}=VD^{t}V^{-1}c}"></span>, where <i>V</i> and <i>D</i> are the matrices formed from the eigenvectors and eigenvalues of <i>A</i>. Since <i>D</i> is diagonal, raising it to power <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 D^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8340d5442a8f73c6137660ca413ec2ef96d64f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle D^{t}}"></span>, just involves raising each element on the diagonal to the power <i>t</i>. This is much easier to do and understand than raising <i>A</i> to power <i>t</i>, since <i>A</i> is usually not diagonal.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Jordan_decomposition">Jordan decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=13" title="Edit section: Jordan decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a> and the <a href="/wiki/Jordan%E2%80%93Chevalley_decomposition" title="Jordan–Chevalley decomposition">Jordan–Chevalley decomposition</a> </p> <ul><li>Applicable to: <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> <i>A</i></li> <li>Comment: the Jordan normal form generalizes the eigendecomposition to cases where there are repeated eigenvalues and cannot be diagonalized, the Jordan–Chevalley decomposition does this without choosing a basis.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Schur_decomposition">Schur decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=14" title="Edit section: Schur decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Schur_decomposition" title="Schur decomposition">Schur decomposition</a></div> <ul><li>Applicable to: <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> <i>A</i></li> <li>Decomposition (complex version): <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=UTU^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>U</mi> <mi>T</mi> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=UTU^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914ec778cd8892ca2579443f798ee94e07f25e8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.156ex; height:2.343ex;" alt="{\displaystyle A=UTU^{*}}"></span>, where <i>U</i> is a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>, <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 U^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8def9b6d2f0acfca452861c11dc2a08419765efe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.895ex; height:2.343ex;" alt="{\displaystyle U^{*}}"></span> is the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of <i>U</i>, and <i>T</i> is an <a href="/wiki/Upper_triangular" class="mw-redirect" title="Upper triangular">upper triangular</a> matrix called the complex <a href="/wiki/Schur_form" class="mw-redirect" title="Schur form">Schur form</a> which has the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of <i>A</i> along its diagonal.</li> <li>Comment: if <i>A</i> is a <a href="/wiki/Normal_matrix" title="Normal matrix">normal matrix</a>, then <i>T</i> is diagonal and the Schur decomposition coincides with the spectral decomposition.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Real_Schur_decomposition">Real Schur decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=15" title="Edit section: Real Schur decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Applicable to: <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> <i>A</i></li> <li>Decomposition: This is a version of Schur decomposition 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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> and <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"> <mi>S</mi> </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/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> only contain real numbers. One can always write <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=VSV^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>V</mi> <mi>S</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=VSV^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/035cf7b294191a081135d777e4e2f5e240f78e55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.396ex; height:2.676ex;" alt="{\displaystyle A=VSV^{\mathsf {T}}}"></span> where <i>V</i> is a real <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrix</a>, <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 V^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bd6e850e373d2333763bdcc535186982a1832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.268ex; height:2.676ex;" alt="{\displaystyle V^{\mathsf {T}}}"></span> is the <a href="/wiki/Matrix_transpose" class="mw-redirect" title="Matrix transpose">transpose</a> of <i>V</i>, and <i>S</i> is a <a href="/wiki/Block_matrix" title="Block matrix">block upper triangular</a> matrix called the real <a href="/wiki/Schur_form" class="mw-redirect" title="Schur form">Schur form</a>. The blocks on the diagonal of <i>S</i> are of size 1×1 (in which case they represent real eigenvalues) or 2×2 (in which case they are derived from <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> eigenvalue pairs).</li></ul> <div class="mw-heading mw-heading3"><h3 id="QZ_decomposition">QZ decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=16" title="Edit section: QZ decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/QZ_decomposition" class="mw-redirect" title="QZ decomposition">QZ decomposition</a></div> <ul><li>Also called: <i>generalized Schur decomposition</i></li> <li>Applicable to: <a href="/wiki/Square_matrix" title="Square matrix">square matrices</a> <i>A</i> and <i>B</i></li> <li>Comment: there are two versions of this decomposition: complex and real.</li> <li>Decomposition (complex version): <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=QSZ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>Q</mi> <mi>S</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=QSZ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b844963c492eda3024403ab9406947a351b6857f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.942ex; height:2.676ex;" alt="{\displaystyle A=QSZ^{*}}"></span> and <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 B=QTZ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mi>Q</mi> <mi>T</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=QTZ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc8a60550df19cd526e065cd77a471e4ddd5c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.1ex; height:2.676ex;" alt="{\displaystyle B=QTZ^{*}}"></span> where <i>Q</i> and <i>Z</i> are <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrices</a>, the * superscript represents <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>, and <i>S</i> and <i>T</i> are <a href="/wiki/Upper_triangular" class="mw-redirect" title="Upper triangular">upper triangular</a> matrices.</li> <li>Comment: in the complex QZ decomposition, the ratios of the diagonal elements of <i>S</i> to the corresponding diagonal elements of <i>T</i>, <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 \lambda _{i}=S_{ii}/T_{ii}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}=S_{ii}/T_{ii}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9809604f643fcd96f98523bbb1c42c925d3dac5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.932ex; height:2.843ex;" alt="{\displaystyle \lambda _{i}=S_{ii}/T_{ii}}"></span>, are the generalized <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> that solve the <a href="/wiki/Eigendecomposition_of_a_matrix#Additional_topics" title="Eigendecomposition of a matrix">generalized eigenvalue problem</a> <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\mathbf {v} =\lambda B\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {v} =\lambda B\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0b8a12671a7ed922e60dd4d82ef570ce855158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.783ex; height:2.176ex;" alt="{\displaystyle A\mathbf {v} =\lambda B\mathbf {v} }"></span> (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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" 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="{\displaystyle \lambda }"></span> is an unknown scalar and <b>v</b> is an unknown nonzero vector).</li> <li>Decomposition (real version): <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=QSZ^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>Q</mi> <mi>S</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=QSZ^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eab79ebb6c64b0cbcef15a20d69289991fb69d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.239ex; height:3.009ex;" alt="{\displaystyle A=QSZ^{\mathsf {T}}}"></span> and <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 B=QTZ^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mi>Q</mi> <mi>T</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=QTZ^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbdc9f883194697d09670b77e4c6ece2fb4bae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.397ex; height:3.009ex;" alt="{\displaystyle B=QTZ^{\mathsf {T}}}"></span> where <i>A</i>, <i>B</i>, <i>Q</i>, <i>Z</i>, <i>S</i>, and <i>T</i> are matrices containing real numbers only. In this case <i>Q</i> and <i>Z</i> are <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrices</a>, the <i>T</i> superscript represents <a href="/wiki/Matrix_transpose" class="mw-redirect" title="Matrix transpose">transposition</a>, and <i>S</i> and <i>T</i> are <a href="/wiki/Block_matrix" title="Block matrix">block upper triangular</a> matrices. The blocks on the diagonal of <i>S</i> and <i>T</i> are of size 1×1 or 2×2.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Takagi's_factorization"><span id="Takagi.27s_factorization"></span>Takagi's factorization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=17" title="Edit section: Takagi's factorization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Applicable to: square, complex, symmetric matrix <i>A</i>.</li> <li>Decomposition: <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=VDV^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>V</mi> <mi>D</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=VDV^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47cce0dc345948e787d06dccbd8d5ff7b7b9631f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.822ex; height:2.676ex;" alt="{\displaystyle A=VDV^{\mathsf {T}}}"></span>, where <i>D</i> is a real nonnegative <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a>, and <i>V</i> is <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary</a>. <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 V^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bd6e850e373d2333763bdcc535186982a1832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.268ex; height:2.676ex;" alt="{\displaystyle V^{\mathsf {T}}}"></span> denotes the <a href="/wiki/Matrix_transpose" class="mw-redirect" title="Matrix transpose">matrix transpose</a> of <i>V</i>.</li> <li>Comment: The diagonal elements of <i>D</i> are the nonnegative square roots of the eigenvalues of <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 AA^{*}=VD^{2}V^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>V</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA^{*}=VD^{2}V^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3cf63d43bcde7f50582748f31bd90cbc12ad28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.654ex; height:2.676ex;" alt="{\displaystyle AA^{*}=VD^{2}V^{-1}}"></span>.</li> <li>Comment: <i>V</i> may be complex even if <i>A</i> is real.</li> <li>Comment: This is not a special case of the eigendecomposition (see above), which uses <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 V^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c4ceda21bb312bbd859275eab9b78fe7af8814b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.25ex; height:2.676ex;" alt="{\displaystyle V^{-1}}"></span> instead of <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 V^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bd6e850e373d2333763bdcc535186982a1832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.268ex; height:2.676ex;" alt="{\displaystyle V^{\mathsf {T}}}"></span>. Moreover, if <i>A</i> is not real, it is not Hermitian and the form using <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 V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}"></span> also does not apply.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Singular_value_decomposition">Singular value decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=18" title="Edit section: Singular value decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></div> <ul><li>Applicable to: <i>m</i>-by-<i>n</i> matrix <i>A</i>.</li> <li>Decomposition: <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=UDV^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>U</mi> <mi>D</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=UDV^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/531a636aa65f3deec7a70335af56da06ae056737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.52ex; height:2.343ex;" alt="{\displaystyle A=UDV^{*}}"></span>, where <i>D</i> is a nonnegative <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a>, and <i>U</i> and <i>V</i> 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 U^{*}U=I,V^{*}V=I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>U</mi> <mo>=</mo> <mi>I</mi> <mo>,</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>V</mi> <mo>=</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{*}U=I,V^{*}V=I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d5504e8c68db465ec4f965c18eeb20a1ce4aa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.011ex; height:2.676ex;" alt="{\displaystyle U^{*}U=I,V^{*}V=I}"></span>. Here <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 V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}"></span> is the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of <i>V</i> (or simply the <a href="/wiki/Matrix_transpose" class="mw-redirect" title="Matrix transpose">transpose</a>, if <i>V</i> contains real numbers only), and <i>I</i> denotes the identity matrix (of some dimension).</li> <li>Comment: The diagonal elements of <i>D</i> are called the <a href="/wiki/Singular_value" title="Singular value">singular values</a> of <i>A</i>.</li> <li>Comment: Like the eigendecomposition above, the singular value decomposition involves finding basis directions along which matrix multiplication is equivalent to scalar multiplication, but it has greater generality since the matrix under consideration need not be square.</li> <li>Uniqueness: the singular values of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> are always uniquely determined. <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> and <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> need not to be unique in general.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Scale-invariant_decompositions">Scale-invariant decompositions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=19" title="Edit section: Scale-invariant decompositions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Refers to variants of existing matrix decompositions, such as the SVD, that are invariant with respect to diagonal scaling. </p> <ul><li>Applicable to: <i>m</i>-by-<i>n</i> matrix <i>A</i>.</li> <li>Unit-Scale-Invariant Singular-Value Decomposition: <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=DUSV^{*}E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>D</mi> <mi>U</mi> <mi>S</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=DUSV^{*}E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06a2371209e3be0521abd0ba378459165c71c9d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.794ex; height:2.343ex;" alt="{\displaystyle A=DUSV^{*}E}"></span>, where <i>S</i> is a unique nonnegative <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> of scale-invariant singular values, <i>U</i> and <i>V</i> are <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrices</a>, <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 V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}"></span> is the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of <i>V</i>, and positive diagonal matrices <i>D</i> and <i>E</i>.</li> <li>Comment: Is analogous to the SVD except that the diagonal elements of <i>S</i> are invariant with respect to left and/or right multiplication of <i>A</i> by arbitrary nonsingular diagonal matrices, as opposed to the standard SVD for which the singular values are invariant with respect to left and/or right multiplication of <i>A</i> by arbitrary unitary matrices.</li> <li>Comment: Is an alternative to the standard SVD when invariance is required with respect to diagonal rather than unitary transformations of <i>A</i>.</li> <li>Uniqueness: The scale-invariant singular values of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> (given by the diagonal elements of <i>S</i>) are always uniquely determined. Diagonal matrices <i>D</i> and <i>E</i>, and unitary <i>U</i> and <i>V</i>, are not necessarily unique in general.</li> <li>Comment: <i>U</i> and <i>V</i> matrices are not the same as those from the SVD.</li></ul> <p>Analogous scale-invariant decompositions can be derived from other matrix decompositions; for example, to obtain scale-invariant eigenvalues.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hessenberg_decomposition">Hessenberg decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=20" title="Edit section: Hessenberg decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Applicable to: <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> A.</li> <li>Decomposition: <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=PHP^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>P</mi> <mi>H</mi> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=PHP^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/050e33f81faf6620cc671fb12427d54951b97858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.526ex; height:2.343ex;" alt="{\displaystyle A=PHP^{*}}"></span> 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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is the <a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg matrix</a> and <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>.</li> <li>Comment: often the first step in the Schur decomposition.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Complete_orthogonal_decomposition">Complete orthogonal decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=21" title="Edit section: Complete orthogonal decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Complete_orthogonal_decomposition" title="Complete orthogonal decomposition">Complete orthogonal decomposition</a></div> <ul><li>Also known as: <i>UTV decomposition</i>, <i>ULV decomposition</i>, <i>URV decomposition</i>.</li> <li>Applicable to: <i>m</i>-by-<i>n</i> matrix <i>A</i>.</li> <li>Decomposition: <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=UTV^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>U</mi> <mi>T</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=UTV^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5761263cc58e5884b22b4f8aa79e2e3e7cb550e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.232ex; height:2.343ex;" alt="{\displaystyle A=UTV^{*}}"></span>, where <i>T</i> is a <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular matrix</a>, and <i>U</i> and <i>V</i> are <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrices</a>.</li> <li>Comment: Similar to the singular value decomposition and to the Schur decomposition.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Other_decompositions">Other decompositions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=22" title="Edit section: Other decompositions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Polar_decomposition">Polar decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=23" title="Edit section: Polar decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></div> <ul><li>Applicable to: any square complex matrix <i>A</i>.</li> <li>Decomposition: <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=UP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>U</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=UP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e8b6f27cda1bc84949a7086953e5fe7b27a1851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.37ex; height:2.176ex;" alt="{\displaystyle A=UP}"></span> (right polar decomposition) or <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=P'U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>P</mi> <mo>′</mo> </msup> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=P'U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0771fe74181ea45247438d7dad0cf93d409268d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.13ex; height:2.509ex;" alt="{\displaystyle A=P'U}"></span> (left polar decomposition), where <i>U</i> is a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a> and <i>P</i> and <i>P'</i> are <a href="/wiki/Positive_semidefinite_matrix" class="mw-redirect" title="Positive semidefinite matrix">positive semidefinite</a> <a href="/wiki/Hermitian_matrices" class="mw-redirect" title="Hermitian matrices">Hermitian matrices</a>.</li> <li>Uniqueness: <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is always unique and equal to <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 {\sqrt {A^{*}A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>A</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {A^{*}A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f59b836838e64c5187d2062ab292b0db876dcf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.476ex; height:3.009ex;" alt="{\displaystyle {\sqrt {A^{*}A}}}"></span> (which is always hermitian and positive semidefinite). If <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is invertible, then <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is unique.</li> <li>Comment: Since any Hermitian matrix admits a spectral decomposition with a unitary 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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> can be written as <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 P=VDV^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>V</mi> <mi>D</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=VDV^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee54661bd60e9068230cea018af2c28922d1328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.527ex; height:2.343ex;" alt="{\displaystyle P=VDV^{*}}"></span>. Since <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is positive semidefinite, all elements in <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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> are non-negative. Since the product of two unitary matrices is unitary, taking <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 W=UV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>U</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=UV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a81f0b1c43f8f541040679b50fb86ba07a47f5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.104ex; height:2.176ex;" alt="{\displaystyle W=UV}"></span>one can write <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=U(VDV^{*})=WDV^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mi>D</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>W</mi> <mi>D</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=U(VDV^{*})=WDV^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921b407e252ca7f23c2d5418935712dbdb0c9f36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.545ex; height:2.843ex;" alt="{\displaystyle A=U(VDV^{*})=WDV^{*}}"></span> which is the singular value decomposition. Hence, the existence of the polar decomposition is equivalent to the existence of the singular value decomposition.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Algebraic_polar_decomposition">Algebraic polar decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=24" title="Edit section: Algebraic polar decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Applicable to: square, complex, non-singular matrix <i>A</i>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>Decomposition: <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=QS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>Q</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=QS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a83c5f700d1833fb8d1718223e31d18d318bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.179ex; height:2.509ex;" alt="{\displaystyle A=QS}"></span>, where <i>Q</i> is a complex orthogonal matrix and <i>S</i> is complex symmetric matrix.</li> <li>Uniqueness: If <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^{\mathsf {T}}A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mathsf {T}}A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/394cb0610f8e3f0a5511dc610ae5694be3436070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.838ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {T}}A}"></span> has no negative real eigenvalues, then the decomposition is unique.<sup id="cite_ref-:0_7-0" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>Comment: The existence of this decomposition is equivalent to <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 AA^{\mathsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA^{\mathsf {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af621dc24b8fc32577441725e8a8a112508ebcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.838ex; height:2.676ex;" alt="{\displaystyle AA^{\mathsf {T}}}"></span> being similar to <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^{\mathsf {T}}A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mathsf {T}}A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/394cb0610f8e3f0a5511dc610ae5694be3436070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.838ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {T}}A}"></span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li>Comment: A variant of this decomposition is <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=RC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>R</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=RC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/063ba776a56330a73d7c66452886a489c74fbaa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.372ex; height:2.176ex;" alt="{\displaystyle A=RC}"></span>, where <i>R</i> is a real matrix and <i>C</i> is a <a href="/w/index.php?title=Circular_matrix&action=edit&redlink=1" class="new" title="Circular matrix (page does not exist)">circular matrix</a>.<sup id="cite_ref-:0_7-1" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Mostow's_decomposition"><span id="Mostow.27s_decomposition"></span>Mostow's decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=25" title="Edit section: Mostow's decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Applicable to: square, complex, non-singular matrix <i>A</i>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>Decomposition: <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=Ue^{iM}e^{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>U</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>M</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=Ue^{iM}e^{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5bcd0017efc833635abbb9bd0d6760790d8a18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.61ex; height:2.676ex;" alt="{\displaystyle A=Ue^{iM}e^{S}}"></span>, where <i>U</i> is unitary, <i>M</i> is real anti-symmetric and <i>S</i> is real symmetric.</li> <li>Comment: The matrix <i>A</i> can also be decomposed as <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=U_{2}e^{S_{2}}e^{iM_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=U_{2}e^{S_{2}}e^{iM_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af8a3ad08d6072b728194e0d0382d5db586e8f31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.947ex; height:3.009ex;" alt="{\displaystyle A=U_{2}e^{S_{2}}e^{iM_{2}}}"></span>, where <i>U</i><sub>2</sub> is unitary, <i>M</i><sub>2</sub> is real anti-symmetric and <i>S</i><sub>2</sub> is real symmetric.<sup id="cite_ref-:0_7-2" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Sinkhorn_normal_form">Sinkhorn normal form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=26" title="Edit section: Sinkhorn normal form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Sinkhorn%27s_theorem" title="Sinkhorn's theorem">Sinkhorn's theorem</a></div> <ul><li>Applicable to: square real matrix <i>A</i> with strictly positive elements.</li> <li>Decomposition: <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=D_{1}SD_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>S</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=D_{1}SD_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53ee9148c029dfb9d2144145729627bad854cef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.298ex; height:2.509ex;" alt="{\displaystyle A=D_{1}SD_{2}}"></span>, where <i>S</i> is <a href="/wiki/Doubly_stochastic_matrix" title="Doubly stochastic matrix">doubly stochastic</a> and <i>D</i><sub>1</sub> and <i>D</i><sub>2</sub> are real diagonal matrices with strictly positive elements.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Sectoral_decomposition">Sectoral decomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=27" title="Edit section: Sectoral decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Applicable to: square, complex matrix <i>A</i> with <a href="/wiki/Numerical_range" title="Numerical range">numerical range</a> contained in the sector <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_{\alpha }=\left\{re^{i\theta }\in \mathbb {C} \mid r>0,|\theta |\leq \alpha <{\frac {\pi }{2}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>∣</mo> <mi>r</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>α</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\alpha }=\left\{re^{i\theta }\in \mathbb {C} \mid r>0,|\theta |\leq \alpha <{\frac {\pi }{2}}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79449518008c56c90a23ef0b5b0b22adc807c5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.647ex; height:4.843ex;" alt="{\displaystyle S_{\alpha }=\left\{re^{i\theta }\in \mathbb {C} \mid r>0,|\theta |\leq \alpha <{\frac {\pi }{2}}\right\}}"></span>.</li> <li>Decomposition: <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=CZC^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>C</mi> <mi>Z</mi> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=CZC^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641baabf3160b1ffa06e1e2c196bf1352fe540ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.14ex; height:2.343ex;" alt="{\displaystyle A=CZC^{*}}"></span>, where <i>C</i> is an invertible complex matrix and <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 Z=\operatorname {diag} \left(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>diag</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\operatorname {diag} \left(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd5140e1a7abd62381a149654c616c02c05e489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.456ex; height:3.343ex;" alt="{\displaystyle Z=\operatorname {diag} \left(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}}\right)}"></span> with all <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 \left|\theta _{j}\right|\leq \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>|</mo> </mrow> <mo>≤</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\theta _{j}\right|\leq \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7012bc73bca36452c6eb06e224690b697fb0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.88ex; height:3.009ex;" alt="{\displaystyle \left|\theta _{j}\right|\leq \alpha }"></span>.<sup id="cite_ref-Zhang2014_11-0" class="reference"><a href="#cite_note-Zhang2014-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Williamson's_normal_form"><span id="Williamson.27s_normal_form"></span>Williamson's normal form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=28" title="Edit section: Williamson's normal form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Applicable to: square, <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite</a> real matrix <i>A</i> with order 2<i>n</i>×2<i>n</i>.</li> <li>Decomposition: <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=S^{\mathsf {T}}\operatorname {diag} (D,D)S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mi>diag</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>,</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=S^{\mathsf {T}}\operatorname {diag} (D,D)S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e459e20ae39f037f34560878288a9c3b442130a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.557ex; height:3.176ex;" alt="{\displaystyle A=S^{\mathsf {T}}\operatorname {diag} (D,D)S}"></span>, 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 S\in {\text{Sp}}(2n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sp</mtext> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\in {\text{Sp}}(2n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32a8bb2107af55d7004af9fa1565065583c99f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.291ex; height:2.843ex;" alt="{\displaystyle S\in {\text{Sp}}(2n)}"></span> is a <a href="/wiki/Symplectic_matrix" title="Symplectic matrix">symplectic matrix</a> and <i>D</i> is a nonnegative <i>n</i>-by-<i>n</i> diagonal matrix.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Matrix_square_root">Matrix square root</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=29" title="Edit section: Matrix square root"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Square_root_of_a_matrix" title="Square root of a matrix">Square root of a matrix</a></div> <ul><li>Decomposition: <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=BB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>B</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=BB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29f12d63be13cbc5da019d777b986e2d34e05f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.37ex; height:2.176ex;" alt="{\displaystyle A=BB}"></span>, not unique in general.</li> <li>In the case of positive semidefinite <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, there is a unique positive semidefinite <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 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 A=B^{*}B=BB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>B</mi> <mo>=</mo> <mi>B</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=B^{*}B=BB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a47c30d7d37cfac5e24b9a20f5f5696e217288f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.05ex; height:2.343ex;" alt="{\displaystyle A=B^{*}B=BB}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=30" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b> with: examples and additional citations. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Matrix_decomposition&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">December 2014</span>)</i></span></div></td></tr></tbody></table> <p>There exist analogues of the SVD, QR, LU and Cholesky factorizations for <b>quasimatrices</b> and <b>cmatrices</b> or <b>continuous matrices</b>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> A ‘quasimatrix’ is, like a matrix, a rectangular scheme whose elements are indexed, but one discrete index is replaced by a continuous index. Likewise, a ‘cmatrix’, is continuous in both indices. As an example of a cmatrix, one can think of the kernel of an <a href="/wiki/Integral_operator" title="Integral operator">integral operator</a>. </p><p>These factorizations are based on early work by <a href="#CITEREFFredholm1903">Fredholm (1903)</a>, <a href="#CITEREFHilbert1904">Hilbert (1904)</a> and <a href="#CITEREFSchmidt1907">Schmidt (1907)</a>. For an account, and a translation to English of the seminal papers, see <a href="#CITEREFStewart2011">Stewart (2011)</a>. </p> <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=Matrix_decomposition&action=edit&section=31" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Matrix_splitting" title="Matrix splitting">Matrix splitting</a></li> <li><a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">Non-negative matrix factorization</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal component analysis</a></li></ul> <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=Matrix_decomposition&action=edit&section=32" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=33" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">If a non-square matrix is used, however, then the matrix <i>U</i> will also have the same rectangular shape as the original matrix <i>A</i>. And so, calling the matrix <i>U</i> upper triangular would be incorrect as the correct term would be that <i>U</i> is the 'row echelon form' of <i>A</i>. Other than this, there are no differences in LU factorization for square and non-square matrices.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=34" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLay2016" class="citation book cs1">Lay, David C. 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Harlow. p. 142. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-292-09223-2" title="Special:BookSources/978-1-292-09223-2"><bdi>978-1-292-09223-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/920463015">920463015</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+and+its+applications&rft.place=Harlow&rft.pages=142&rft.edition=Fifth+Global&rft.date=2016&rft_id=info%3Aoclcnum%2F920463015&rft.isbn=978-1-292-09223-2&rft.aulast=Lay&rft.aufirst=David+C.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F920463015&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+decomposition" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>)</span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPiziakOdell1999" class="citation journal cs1">Piziak, R.; Odell, P. 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(2018), "A Rank-Preserving Generalized Matrix Inverse for Consistency with Respect to Similarity", <i>IEEE Control Systems Letters</i>, <b>3</b>: <span class="nowrap">91–</span>95, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1804.07334">1804.07334</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FLCSYS.2018.2854240">10.1109/LCSYS.2018.2854240</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/2475-1456">2475-1456</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5031440">5031440</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Control+Systems+Letters&rft.atitle=A+Rank-Preserving+Generalized+Matrix+Inverse+for+Consistency+with+Respect+to+Similarity&rft.volume=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E91-%3C%2Fspan%3E95&rft.date=2018&rft_id=info%3Aarxiv%2F1804.07334&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5031440%23id-name%3DS2CID&rft.issn=2475-1456&rft_id=info%3Adoi%2F10.1109%2FLCSYS.2018.2854240&rft.aulast=Uhlmann&rft.aufirst=J.K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+decomposition" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFChoudhuryHorn1987">Choudhury & Horn 1987</a>, pp. 219–225</span> </li> <li id="cite_note-:0-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_7-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBhatia2013" class="citation journal cs1">Bhatia, Rajendra (2013-11-15). 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Soc., vol. 14, American Mathematical Society, pp. <span class="nowrap">31–</span>54</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Some+new+decomposition+theorems+for+semi-simple+groups&rft.series=Mem.+Amer.+Math.+Soc.&rft.pages=%3Cspan+class%3D%22nowrap%22%3E31-%3C%2Fspan%3E54&rft.pub=American+Mathematical+Society&rft.date=1955&rft.aulast=Mostow&rft.aufirst=G.+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fliealgebrasandli029541mbp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+decomposition" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNielsenBhatia2012" class="citation book cs1">Nielsen, Frank; Bhatia, Rajendra (2012). <i>Matrix Information Geometry</i>. 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(2015), "Continuous analogues of matrix factorizations", <i><a href="/wiki/Proceedings_of_the_Royal_Society" title="Proceedings of the Royal Society">Proc. R. Soc. A</a></i>, <b>471</b> (2173): 20140585, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014RSPSA.47140585T">2014RSPSA.47140585T</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.2014.0585">10.1098/rspa.2014.0585</a>, <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4277194">4277194</a></span>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25568618">25568618</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+R.+Soc.+A&rft.atitle=Continuous+analogues+of+matrix+factorizations&rft.volume=471&rft.issue=2173&rft.pages=20140585&rft.date=2015&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4277194%23id-name%3DPMC&rft_id=info%3Apmid%2F25568618&rft_id=info%3Adoi%2F10.1098%2Frspa.2014.0585&rft_id=info%3Abibcode%2F2014RSPSA.47140585T&rft.aulast=Townsend&rft.aufirst=A.&rft.au=Trefethen%2C+L.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+decomposition" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJun2021" class="citation cs2">Jun, Lu (2021), <i>Numerical matrix decomposition and its modern applications: A rigorous first course</i>, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2107.02579">2107.02579</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+matrix+decomposition+and+its+modern+applications%3A+A+rigorous+first+course&rft.date=2021&rft_id=info%3Aarxiv%2F2107.02579&rft.aulast=Jun&rft.aufirst=Lu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMatrix+decomposition" class="Z3988"></span></li></ul> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrix_decomposition&action=edit&section=36" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.bluebit.gr/matrix-calculator/">Online Matrix Calculator</a></li> <li><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=matrix+decomposition&rawformassumption={%22C%22,+%22matrix+decomposition%22}+-%3E+{%22Calculator%22}&rawformassumption={%22MC%22,%22%22}-%3E{%22Formula%22}">Wolfram Alpha Matrix Decomposition Computation » LU and QR Decomposition</a></li> <li><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Matrix_factorization">Springer Encyclopaedia of Mathematics » Matrix factorization</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110314171151/http://www.graphlab.ml.cmu.edu/pmf.html">GraphLab</a> <a href="/wiki/GraphLab" title="GraphLab">GraphLab</a> <a href="/wiki/Collaborative_filtering" title="Collaborative filtering">collaborative filtering</a> library, large scale parallel implementation of matrix decomposition methods (in C++) for multicore.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist 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