Triangular matrix - Wikipedia

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id="toc-Forward_substitution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Forward_substitution"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Forward substitution</span> </div> </a> <ul id="toc-Forward_substitution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_forms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Special_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Special forms</span> </div> </a> <button aria-controls="toc-Special_forms-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 Special forms subsection</span> </button> <ul id="toc-Special_forms-sublist" class="vector-toc-list"> <li id="toc-Unitriangular_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unitriangular_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Unitriangular matrix</span> </div> </a> <ul id="toc-Unitriangular_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Strictly_triangular_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Strictly_triangular_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Strictly triangular matrix</span> </div> </a> <ul id="toc-Strictly_triangular_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Atomic_triangular_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Atomic_triangular_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Atomic triangular matrix</span> </div> </a> <ul id="toc-Atomic_triangular_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Block_triangular_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Block_triangular_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Block triangular matrix</span> </div> </a> <ul id="toc-Block_triangular_matrix-sublist" class="vector-toc-list"> <li id="toc-Upper_block_triangular" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Upper_block_triangular"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1</span> <span>Upper block triangular</span> </div> </a> <ul id="toc-Upper_block_triangular-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lower_block_triangular" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lower_block_triangular"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.2</span> <span>Lower block triangular</span> </div> </a> <ul id="toc-Lower_block_triangular-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Triangularisability" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Triangularisability"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Triangularisability</span> </div> </a> <button aria-controls="toc-Triangularisability-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 Triangularisability subsection</span> </button> <ul id="toc-Triangularisability-sublist" class="vector-toc-list"> <li id="toc-Simultaneous_triangularisability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simultaneous_triangularisability"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Simultaneous triangularisability</span> </div> </a> <ul id="toc-Simultaneous_triangularisability-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algebras_of_triangular_matrices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebras_of_triangular_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Algebras of triangular matrices</span> </div> </a> <button aria-controls="toc-Algebras_of_triangular_matrices-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 Algebras of triangular matrices subsection</span> </button> <ul id="toc-Algebras_of_triangular_matrices-sublist" class="vector-toc-list"> <li id="toc-Borel_subgroups_and_Borel_subalgebras" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Borel_subgroups_and_Borel_subalgebras"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Borel subgroups and Borel subalgebras</span> </div> </a> <ul id="toc-Borel_subgroups_and_Borel_subalgebras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Examples</span> </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <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">Triangular matrix</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 36 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-36" 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">36 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/%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9_%D9%85%D8%AB%D9%84%D8%AB%D9%8A%D8%A9" title="مصفوفة مثلثية – Arabic" lang="ar" hreflang="ar" data-title="مصفوفة مثلثية" 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/Matriu_triangular" title="Matriu triangular – Catalan" lang="ca" hreflang="ca" data-title="Matriu triangular" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Troj%C3%BAheln%C3%ADkov%C3%A1_matice" title="Trojúhelníková matice – Czech" lang="cs" hreflang="cs" data-title="Trojúhelníková matice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dreiecksmatrix" title="Dreiecksmatrix – German" lang="de" hreflang="de" data-title="Dreiecksmatrix" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kolmnurkmaatriks" title="Kolmnurkmaatriks – Estonian" lang="et" hreflang="et" data-title="Kolmnurkmaatriks" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%81%CE%B9%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CF%8C%CF%82_%CF%80%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82" title="Τριγωνικός πίνακας – Greek" lang="el" hreflang="el" data-title="Τριγωνικός πίνακας" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matriz_triangular" title="Matriz triangular – Spanish" lang="es" hreflang="es" data-title="Matriz triangular" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Triangula_matrico" title="Triangula matrico – Esperanto" lang="eo" hreflang="eo" data-title="Triangula matrico" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Matrize_triangeluar" title="Matrize triangeluar – Basque" lang="eu" hreflang="eu" data-title="Matrize triangeluar" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3_%D9%85%D8%AB%D9%84%D8%AB%DB%8C" title="ماتریس مثلثی – Persian" lang="fa" hreflang="fa" data-title="ماتریس مثلثی" 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/Matrice_triangulaire" title="Matrice triangulaire – French" lang="fr" hreflang="fr" data-title="Matrice triangulaire" 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/Matriz_triangular" title="Matriz triangular – Galician" lang="gl" hreflang="gl" data-title="Matriz triangular" 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/%EC%82%BC%EA%B0%81%ED%96%89%EB%A0%AC" title="삼각행렬 – Korean" lang="ko" hreflang="ko" data-title="삼각행렬" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matriks_segitiga" title="Matriks segitiga – Indonesian" lang="id" hreflang="id" data-title="Matriks segitiga" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%9Er%C3%ADhyrningsfylki" title="Þríhyrningsfylki – Icelandic" lang="is" hreflang="is" data-title="Þríhyrningsfylki" 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/Matrice_triangolare" title="Matrice triangolare – Italian" lang="it" hreflang="it" data-title="Matrice triangolare" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%9E%D7%A9%D7%95%D7%9C%D7%A9%D7%99%D7%AA" title="מטריצה משולשית – Hebrew" lang="he" hreflang="he" data-title="מטריצה משולשית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Matris_triang%C3%BClara" title="Matris triangülara – Lombard" lang="lmo" hreflang="lmo" data-title="Matris triangülara" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/H%C3%A1romsz%C3%B6gm%C3%A1trix" title="Háromszögmátrix – Hungarian" lang="hu" hreflang="hu" data-title="Háromszögmátrix" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Driehoeksmatrix" title="Driehoeksmatrix – Dutch" lang="nl" hreflang="nl" data-title="Driehoeksmatrix" 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/%E4%B8%89%E8%A7%92%E8%A1%8C%E5%88%97" title="三角行列 – Japanese" lang="ja" hreflang="ja" data-title="三角行列" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Triangul%C3%A6r_matrise" title="Triangulær matrise – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Triangulær matrise" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Triangul%C3%A6r_matrise" title="Triangulær matrise – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Triangulær matrise" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9A%D1%83%D0%BC%D0%BB%D1%83%D0%BA%D0%B0%D0%BD_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B5" title="Кумлукан матрице – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Кумлукан матрице" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Macierz_tr%C3%B3jk%C4%85tna" title="Macierz trójkątna – Polish" lang="pl" hreflang="pl" data-title="Macierz trójkątna" 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/Matriz_triangular" title="Matriz triangular – Portuguese" lang="pt" hreflang="pt" data-title="Matriz triangular" 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%A2%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Треугольная матрица – Russian" lang="ru" hreflang="ru" data-title="Треугольная матрица" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Trikotna_matrika" title="Trikotna matrika – Slovenian" lang="sl" hreflang="sl" data-title="Trikotna matrika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kolmiomatriisi" title="Kolmiomatriisi – Finnish" lang="fi" hreflang="fi" data-title="Kolmiomatriisi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Triangul%C3%A4r_matris" title="Triangulär matris – Swedish" lang="sv" hreflang="sv" data-title="Triangulär matris" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%A3_%E0%AE%85%E0%AE%A3%E0%AE%BF" title="முக்கோண அணி – Tamil" lang="ta" hreflang="ta" data-title="முக்கோண அணி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%9C%C3%A7gen_matris" title="Üçgen matris – Turkish" lang="tr" hreflang="tr" data-title="Üçgen matris" 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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 <a href="/wiki/Triangular_array" title="Triangular array">triangular array</a>, a related concept.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For the rings, see <a href="/wiki/Triangular_matrix_ring" title="Triangular matrix ring">triangular matrix ring</a>.</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Triangularization" redirects here. For the geometric process, see <a href="/wiki/Triangulation" title="Triangulation">Triangulation</a>.</div> <p>In mathematics, a <b>triangular matrix</b> is a special kind of <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a>. A square matrix is called <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="lower_triangular"></span><span class="vanchor-text">lower triangular</span></span></b> if all the entries <i>above</i> the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> are zero. Similarly, a square matrix is called <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="upper_triangular"></span><span class="vanchor-text">upper triangular</span></span></b> if all the entries <i>below</i> the main diagonal are zero. </p><p>Because matrix equations with triangular matrices are easier to solve, they are very important in <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>. By the <a href="/wiki/LU_decomposition" title="LU decomposition">LU decomposition</a> algorithm, an <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible matrix</a> may be written as the <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">product</a> of a lower triangular matrix <i>L</i> and an upper triangular matrix <i>U</i> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> all its leading principal <a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">minors</a> are non-zero. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A matrix of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\begin{bmatrix}\ell _{1,1}&&&&0\\\ell _{2,1}&\ell _{2,2}&&&\\\ell _{3,1}&\ell _{3,2}&\ddots &&\\\vdots &\vdots &\ddots &\ddots &\\\ell _{n,1}&\ell _{n,2}&\ldots &\ell _{n,n-1}&\ell _{n,n}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd /> <mtd /> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd /> <mtd /> <mtd /> </mtr> <mtr> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\begin{bmatrix}\ell _{1,1}&&&&0\\\ell _{2,1}&\ell _{2,2}&&&\\\ell _{3,1}&\ell _{3,2}&\ddots &&\\\vdots &\vdots &\ddots &\ddots &\\\ell _{n,1}&\ell _{n,2}&\ldots &\ell _{n,n-1}&\ell _{n,n}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2b80e747e5205f0f5544260f2943e8e0056b0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:37.102ex; height:20.009ex;" alt="{\displaystyle L={\begin{bmatrix}\ell _{1,1}&&&&0\\\ell _{2,1}&\ell _{2,2}&&&\\\ell _{3,1}&\ell _{3,2}&\ddots &&\\\vdots &\vdots &\ddots &\ddots &\\\ell _{n,1}&\ell _{n,2}&\ldots &\ell _{n,n-1}&\ell _{n,n}\end{bmatrix}}}"></span></dd></dl> <p>is called a <b>lower triangular matrix</b> or <b>left triangular matrix</b>, and analogously a matrix of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U={\begin{bmatrix}u_{1,1}&u_{1,2}&u_{1,3}&\ldots &u_{1,n}\\&u_{2,2}&u_{2,3}&\ldots &u_{2,n}\\&&\ddots &\ddots &\vdots \\&&&\ddots &u_{n-1,n}\\0&&&&u_{n,n}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd /> <mtd /> <mtd /> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U={\begin{bmatrix}u_{1,1}&u_{1,2}&u_{1,3}&\ldots &u_{1,n}\\&u_{2,2}&u_{2,3}&\ldots &u_{2,n}\\&&\ddots &\ddots &\vdots \\&&&\ddots &u_{n-1,n}\\0&&&&u_{n,n}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f770c7fa4c212eac3d7d7f9a54f7decbc811276f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:38.084ex; height:20.009ex;" alt="{\displaystyle U={\begin{bmatrix}u_{1,1}&u_{1,2}&u_{1,3}&\ldots &u_{1,n}\\&u_{2,2}&u_{2,3}&\ldots &u_{2,n}\\&&\ddots &\ddots &\vdots \\&&&\ddots &u_{n-1,n}\\0&&&&u_{n,n}\end{bmatrix}}}"></span></dd></dl> <p>is called an <b>upper triangular matrix</b> or <b>right triangular matrix</b>. A lower or left triangular matrix is commonly denoted with the variable <i>L</i>, and an upper or right triangular matrix is commonly denoted with the variable <i>U</i> or <i>R</i>. </p><p>A matrix that is both upper and lower triangular is <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal</a>. Matrices that are <a href="/wiki/Similar_(linear_algebra)" class="mw-redirect" title="Similar (linear algebra)">similar</a> to triangular matrices are called <b>triangularisable</b>. </p><p>A non-square (or sometimes any) matrix with zeros above (below) the diagonal is called a lower (upper) trapezoidal matrix. The non-zero entries form the shape of a <a href="/wiki/Trapezoid" title="Trapezoid">trapezoid</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The matrix </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&0\\2&96&0\\4&9&69\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>96</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>69</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&0\\2&96&0\\4&9&69\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf6b5b1167126393531f3195d7b2830d614f47b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.31ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0&0\\2&96&0\\4&9&69\end{bmatrix}}}"></span></dd></dl> <p>is lower triangular, and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&4&1\\0&6&9\\0&0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>9</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&4&1\\0&6&9\\0&0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f2acefe00f7f13068c53271513f901d844b977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.985ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&4&1\\0&6&9\\0&0&1\end{bmatrix}}}"></span></dd></dl> <p>is upper triangular. </p> <div class="mw-heading mw-heading2"><h2 id="Forward_and_back_substitution">Forward and back substitution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=3" title="Edit section: Forward and back substitution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A matrix equation in 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 L\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</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 L\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff55e736bb4c950832d352913df1938dd28a3bce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.578ex; height:2.176ex;" alt="{\displaystyle L\mathbf {x} =\mathbf {b} }"></span> 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 U\mathbf {x} =\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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ecbcc9050691826631136244038b861b2f4e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.777ex; height:2.176ex;" alt="{\displaystyle U\mathbf {x} =\mathbf {b} }"></span> is very easy to solve by an iterative process called <b>forward substitution</b> for lower triangular matrices and analogously <b>back substitution</b> for upper triangular matrices. The process is so called because for lower triangular matrices, one first computes <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span>, then substitutes that <i>forward</i> into the <i>next</i> equation to solve for <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}"></span>, and repeats through 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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"></span>. In an upper triangular matrix, one works <i>backwards,</i> first computing <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}"></span>, then substituting that <i>back</i> into the <i>previous</i> equation to solve for <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_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56ef8a744e0bb10d512f2e77625811bff9771d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.649ex; height:2.009ex;" alt="{\displaystyle x_{n-1}}"></span>, and repeating through <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span>. </p><p>Notice that this does not require inverting the matrix. </p> <div class="mw-heading mw-heading3"><h3 id="Forward_substitution">Forward substitution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=4" title="Edit section: Forward substitution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The matrix equation <i>L</i><b>x</b> = <b>b</b> can be written as a system of linear equations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\ell _{1,1}x_{1}&&&&&&&=&b_{1}\\\ell _{2,1}x_{1}&+&\ell _{2,2}x_{2}&&&&&=&b_{2}\\\vdots &&\vdots &&\ddots &&&&\vdots \\\ell _{m,1}x_{1}&+&\ell _{m,2}x_{2}&+&\dotsb &+&\ell _{m,m}x_{m}&=&b_{m}\\\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd /> <mtd /> <mtd /> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd /> <mtd /> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo>+</mo> </mtd> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\ell _{1,1}x_{1}&&&&&&&=&b_{1}\\\ell _{2,1}x_{1}&+&\ell _{2,2}x_{2}&&&&&=&b_{2}\\\vdots &&\vdots &&\ddots &&&&\vdots \\\ell _{m,1}x_{1}&+&\ell _{m,2}x_{2}&+&\dotsb &+&\ell _{m,m}x_{m}&=&b_{m}\\\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c466811e460b9348543167d352f5f12d6723695e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:52.382ex; height:14.843ex;" alt="{\displaystyle {\begin{matrix}\ell _{1,1}x_{1}&&&&&&&=&b_{1}\\\ell _{2,1}x_{1}&+&\ell _{2,2}x_{2}&&&&&=&b_{2}\\\vdots &&\vdots &&\ddots &&&&\vdots \\\ell _{m,1}x_{1}&+&\ell _{m,2}x_{2}&+&\dotsb &+&\ell _{m,m}x_{m}&=&b_{m}\\\end{matrix}}}"></span></dd></dl> <p>Observe that the first 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 \ell _{1,1}x_{1}=b_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{1,1}x_{1}=b_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20c11da4a6a22cb692424b1fa85406368f16617a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.837ex; height:2.843ex;" alt="{\displaystyle \ell _{1,1}x_{1}=b_{1}}"></span>) only involves <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span>, and thus one can solve for <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> directly. The second equation only involves <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></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 x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}"></span>, and thus can be solved once one substitutes in the already solved value for <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span>. Continuing in this way, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>-th equation only involves <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_{1},\dots ,x_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\dots ,x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49099bbc969b384b05477fd616862198234d9d5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.981ex; height:2.009ex;" alt="{\displaystyle x_{1},\dots ,x_{k}}"></span>, and one can solve for <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_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2b88c64c76a03611549fb9b4cf4ed060b56002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}"></span> using the previously solved values for <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_{1},\dots ,x_{k-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\dots ,x_{k-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece24a6913d611d8804774462d8e5edcf87cb979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.081ex; height:2.009ex;" alt="{\displaystyle x_{1},\dots ,x_{k-1}}"></span>. The resulting formulas are: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x_{1}&={\frac {b_{1}}{\ell _{1,1}}},\\x_{2}&={\frac {b_{2}-\ell _{2,1}x_{1}}{\ell _{2,2}}},\\&\ \ \vdots \\x_{m}&={\frac {b_{m}-\sum _{i=1}^{m-1}\ell _{m,i}x_{i}}{\ell _{m,m}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mtext> </mtext> <mtext> </mtext> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{1}&={\frac {b_{1}}{\ell _{1,1}}},\\x_{2}&={\frac {b_{2}-\ell _{2,1}x_{1}}{\ell _{2,2}}},\\&\ \ \vdots \\x_{m}&={\frac {b_{m}-\sum _{i=1}^{m-1}\ell _{m,i}x_{i}}{\ell _{m,m}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aeb7423e4c1fd1a4072d63801854ce389ac4cb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.338ex; width:26.266ex; height:23.843ex;" alt="{\displaystyle {\begin{aligned}x_{1}&={\frac {b_{1}}{\ell _{1,1}}},\\x_{2}&={\frac {b_{2}-\ell _{2,1}x_{1}}{\ell _{2,2}}},\\&\ \ \vdots \\x_{m}&={\frac {b_{m}-\sum _{i=1}^{m-1}\ell _{m,i}x_{i}}{\ell _{m,m}}}.\end{aligned}}}"></span></dd></dl> <p>A matrix equation with an upper triangular matrix <i>U</i> can be solved in an analogous way, only working backwards. </p> <div class="mw-heading mw-heading3"><h3 id="Applications">Applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=5" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Forward substitution is used in financial <a href="/wiki/Bootstrapping_(finance)" title="Bootstrapping (finance)">bootstrapping</a> to construct a <a href="/wiki/Yield_curve" title="Yield curve">yield curve</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=6" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Transpose" title="Transpose">transpose</a> of an upper triangular matrix is a lower triangular matrix and vice versa. </p><p>A matrix which is both symmetric and triangular is diagonal. In a similar vein, a matrix which is both <a href="/wiki/Normal_matrix" title="Normal matrix">normal</a> (meaning <i>A</i><sup>*</sup><i>A</i> = <i>AA</i><sup>*</sup>, where <i>A</i><sup>*</sup> is the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>) and triangular is also diagonal. This can be seen by looking at the diagonal entries of <i>A</i><sup>*</sup><i>A</i> and <i>AA</i><sup>*</sup>. </p><p>The <a href="/wiki/Determinant" title="Determinant">determinant</a> and <a href="/wiki/Permanent_(mathematics)" title="Permanent (mathematics)">permanent</a> of a triangular matrix equal the product of the diagonal entries, as can be checked by direct computation. </p><p>In fact more is true: the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of a triangular matrix are exactly its diagonal entries. Moreover, each eigenvalue occurs exactly <i>k</i> times on the diagonal, where <i>k</i> is its <a href="/wiki/Algebraic_multiplicity" class="mw-redirect" title="Algebraic multiplicity">algebraic multiplicity</a>, that is, its <a href="/wiki/Multiplicity_of_a_root_of_a_polynomial" class="mw-redirect" title="Multiplicity of a root of a polynomial">multiplicity as a root</a> of the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</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 p_{A}(x)=\det(xI-A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>I</mi> <mo>−</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{A}(x)=\det(xI-A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/887a166d61effbb14d1eee08249b7e8596a416e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:21.085ex; height:2.843ex;" alt="{\displaystyle p_{A}(x)=\det(xI-A)}"></span> of <i>A</i>. In other words, the characteristic polynomial of a triangular <i>n</i>×<i>n</i> matrix <i>A</i> is exactly </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{A}(x)=(x-a_{11})(x-a_{22})\cdots (x-a_{nn})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{A}(x)=(x-a_{11})(x-a_{22})\cdots (x-a_{nn})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bb628f891030fdec4a8f6bbbe191e95b69ac028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:40.043ex; height:2.843ex;" alt="{\displaystyle p_{A}(x)=(x-a_{11})(x-a_{22})\cdots (x-a_{nn})}"></span>,</dd></dl> <p>that is, the unique degree <i>n</i> polynomial whose roots are the diagonal entries of <i>A</i> (with multiplicities). To see this, observe 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 xI-A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>I</mi> <mo>−</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xI-A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aeb5d21d0cf0aaf43e8a190ddd661f2cc95efb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.085ex; height:2.343ex;" alt="{\displaystyle xI-A}"></span> is also triangular and hence its determinant <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 \det(xI-A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>I</mi> <mo>−</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(xI-A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/681acbc82ab2d91d7ffb90b906ceb646c9ab8d56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.124ex; height:2.843ex;" alt="{\displaystyle \det(xI-A)}"></span> is the product of its diagonal entries <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-a_{11})(x-a_{22})\cdots (x-a_{nn})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-a_{11})(x-a_{22})\cdots (x-a_{nn})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315ab0aad295a774d7a707f3099a6a839282e2d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.082ex; height:2.843ex;" alt="{\displaystyle (x-a_{11})(x-a_{22})\cdots (x-a_{nn})}"></span>.<sup id="cite_ref-axler_1-0" class="reference"><a href="#cite_note-axler-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Special_forms">Special forms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=7" title="Edit section: Special forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Unitriangular_matrix">Unitriangular matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=8" title="Edit section: Unitriangular matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the entries on the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> of a (upper or lower) triangular matrix are all 1, the matrix is called (upper or lower) <b>unitriangular</b>. </p><p>Other names used for these matrices are <b>unit</b> (upper or lower) <b>triangular</b>, or very rarely <b>normed</b> (upper or lower) <b>triangular</b>. However, a <i>unit</i> triangular matrix is not the same as <b>the</b> <i><a href="/wiki/Identity_matrix" title="Identity matrix">unit matrix</a></i>, and a <i>normed</i> triangular matrix has nothing to do with the notion of <a href="/wiki/Matrix_norm" title="Matrix norm">matrix norm</a>. </p><p>All finite unitriangular matrices are <a href="/wiki/Unipotent" title="Unipotent">unipotent</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Strictly_triangular_matrix">Strictly triangular matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=9" title="Edit section: Strictly triangular matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If all of the entries on the main diagonal of a (upper or lower) triangular matrix are also 0, the matrix is called <b>strictly</b> (upper or lower) <b>triangular</b>. </p><p>All finite strictly triangular matrices are <a href="/wiki/Nilpotent_matrix" title="Nilpotent matrix">nilpotent</a> of index at most <i>n</i> as a consequence of the <a href="/wiki/Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley-Hamilton theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Atomic_triangular_matrix">Atomic triangular matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=10" title="Edit section: Atomic triangular matrix"><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/Frobenius_matrix" title="Frobenius matrix">Frobenius matrix</a></div> <p>An <b>atomic</b> (upper or lower) <b>triangular matrix</b> is a special form of unitriangular matrix, where all of the <a href="/wiki/Off-diagonal_element" class="mw-redirect" title="Off-diagonal element">off-diagonal elements</a> are zero, except for the entries in a single column. Such a matrix is also called a <b>Frobenius matrix</b>, a <b>Gauss matrix</b>, or a <b>Gauss transformation matrix</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Block_triangular_matrix">Block triangular matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=11" title="Edit section: Block triangular matrix"><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_matrix" title="Block matrix">Block matrix</a></div> <p>A block triangular matrix is a <a href="/wiki/Block_matrix" title="Block matrix">block matrix</a> (partitioned matrix) that is a triangular matrix. </p> <div class="mw-heading mw-heading4"><h4 id="Upper_block_triangular">Upper block triangular</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=12" title="Edit section: Upper block triangular"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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> is <b>upper block triangular</b> if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1k}\\0&A_{22}&\cdots &A_{2k}\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &A_{kk}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1k}\\0&A_{22}&\cdots &A_{2k}\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &A_{kk}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd679e90248b8397597de37d2f4878ee8a20231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:29.567ex; height:14.176ex;" alt="{\displaystyle A={\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1k}\\0&A_{22}&\cdots &A_{2k}\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &A_{kk}\end{bmatrix}}}"></span>,</dd></dl> <p>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_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f84075572137ed9a9a2091c2cd4b103d4d5a6351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.304ex; height:3.009ex;" alt="{\displaystyle A_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}}"></span> for 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 i,j=1,\ldots ,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j=1,\ldots ,k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27cc5829e1059a893bec0e372a4e4d0e83f080c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.445ex; height:2.509ex;" alt="{\displaystyle i,j=1,\ldots ,k}"></span>.<sup id="cite_ref-bernstein2009_2-0" class="reference"><a href="#cite_note-bernstein2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Lower_block_triangular">Lower block triangular</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=13" title="Edit section: Lower block triangular"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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> is <b>lower block triangular</b> if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}A_{11}&0&\cdots &0\\A_{21}&A_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\A_{k1}&A_{k2}&\cdots &A_{kk}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}A_{11}&0&\cdots &0\\A_{21}&A_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\A_{k1}&A_{k2}&\cdots &A_{kk}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ada0092bb921ad6089dc8e4018a336f66c1292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:29.636ex; height:14.176ex;" alt="{\displaystyle A={\begin{bmatrix}A_{11}&0&\cdots &0\\A_{21}&A_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\A_{k1}&A_{k2}&\cdots &A_{kk}\end{bmatrix}}}"></span>,</dd></dl> <p>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_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f84075572137ed9a9a2091c2cd4b103d4d5a6351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.304ex; height:3.009ex;" alt="{\displaystyle A_{ij}\in \mathbb {F} ^{n_{i}\times n_{j}}}"></span> for 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 i,j=1,\ldots ,k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j=1,\ldots ,k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27cc5829e1059a893bec0e372a4e4d0e83f080c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.445ex; height:2.509ex;" alt="{\displaystyle i,j=1,\ldots ,k}"></span>.<sup id="cite_ref-bernstein2009_2-1" class="reference"><a href="#cite_note-bernstein2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Triangularisability">Triangularisability<span class="anchor" id="Triangularizability"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=14" title="Edit section: Triangularisability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A matrix that is <a href="/wiki/Similar_matrix" class="mw-redirect" title="Similar matrix">similar</a> to a triangular matrix is referred to as <b>triangularizable</b>. Abstractly, this is equivalent to stabilizing a <a href="/wiki/Flag_(linear_algebra)" title="Flag (linear algebra)">flag</a>: upper triangular matrices are precisely those that preserve the <a href="/wiki/Standard_flag" class="mw-redirect" title="Standard flag">standard flag</a>, which is given by the standard ordered basis <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 (e_{1},\ldots ,e_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e_{1},\ldots ,e_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12c4e5513493af62a4753b821ba5bc7842a5278a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.427ex; height:2.843ex;" alt="{\displaystyle (e_{1},\ldots ,e_{n})}"></span> and the resulting flag <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 0<\left\langle e_{1}\right\rangle <\left\langle e_{1},e_{2}\right\rangle <\cdots <\left\langle e_{1},\ldots ,e_{n}\right\rangle =K^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mrow> <mo>⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⟩</mo> </mrow> <mo><</mo> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo><</mo> <mo>⋯</mo> <mo><</mo> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\left\langle e_{1}\right\rangle <\left\langle e_{1},e_{2}\right\rangle <\cdots <\left\langle e_{1},\ldots ,e_{n}\right\rangle =K^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/950e759b18d398b691d84f07734eec75bd8d6955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.83ex; height:2.843ex;" alt="{\displaystyle 0<\left\langle e_{1}\right\rangle <\left\langle e_{1},e_{2}\right\rangle <\cdots <\left\langle e_{1},\ldots ,e_{n}\right\rangle =K^{n}.}"></span> All flags are conjugate (as the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> acts transitively on bases), so any matrix that stabilises a flag is similar to one that stabilizes the standard flag. </p><p>Any complex square matrix is triangularizable.<sup id="cite_ref-axler_1-1" class="reference"><a href="#cite_note-axler-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In fact, a matrix <i>A</i> over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> containing all of the eigenvalues of <i>A</i> (for example, any matrix over an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>) is similar to a triangular matrix. This can be proven by using induction on the fact that <i>A</i> has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that <i>A</i> stabilizes a flag, and is thus triangularizable with respect to a basis for that flag. </p><p>A more precise statement is given by the <a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a> theorem, which states that in this situation, <i>A</i> is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.<sup id="cite_ref-axler_1-2" class="reference"><a href="#cite_note-axler-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-herstein_3-0" class="reference"><a href="#cite_note-herstein-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In the case of complex matrices, it is possible to say more about triangularization, namely, that any square matrix <i>A</i> has a <a href="/wiki/Schur_decomposition" title="Schur decomposition">Schur decomposition</a>. This means that <i>A</i> is unitarily equivalent (i.e. similar, using a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a> as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag. </p> <div class="mw-heading mw-heading3"><h3 id="Simultaneous_triangularisability">Simultaneous triangularisability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=15" title="Edit section: Simultaneous triangularisability"><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">See also: <a href="/wiki/Simultaneously_diagonalizable" class="mw-redirect" title="Simultaneously diagonalizable">Simultaneously diagonalizable</a></div> <p>A set of matrices <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_{1},\ldots ,A_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbe8023d2314d4c8322622f36d0dedc3d93a867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.807ex; height:2.509ex;" alt="{\displaystyle A_{1},\ldots ,A_{k}}"></span> are said to be <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="simultaneously_triangularisable"></span><span class="vanchor-text">simultaneously triangularisable</span></span></b> if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix <i>P.</i> Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124dbf41624fcbdaa7fc03abf67bb8566e3d20f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.19ex; height:2.509ex;" alt="{\displaystyle A_{i},}"></span> denoted <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 K[A_{1},\ldots ,A_{k}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[A_{1},\ldots ,A_{k}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e14130e32374bc22b42759ddec8cdaf131ce97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.814ex; height:2.843ex;" alt="{\displaystyle K[A_{1},\ldots ,A_{k}].}"></span> Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a <a href="/wiki/Borel_subalgebra" title="Borel subalgebra">Borel subalgebra</a>. </p><p>The basic result is that (over an algebraically closed field), the <a href="/wiki/Commuting_matrices" title="Commuting matrices">commuting 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 A,B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}"></span> or more generally <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_{1},\ldots ,A_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbe8023d2314d4c8322622f36d0dedc3d93a867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.807ex; height:2.509ex;" alt="{\displaystyle A_{1},\ldots ,A_{k}}"></span> are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. This was proven by Frobenius, starting in 1878 for a commuting pair, as discussed at <a href="/wiki/Commuting_matrices" title="Commuting matrices">commuting matrices</a>. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices. </p><p>The fact that commuting matrices have a common eigenvector can be interpreted as a result of <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a>: commuting matrices form a commutative algebra <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 K[A_{1},\ldots ,A_{k}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[A_{1},\ldots ,A_{k}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4fc023b03e3af38fd10c87c2eab8611e50b3e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.167ex; height:2.843ex;" alt="{\displaystyle K[A_{1},\ldots ,A_{k}]}"></span> over <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 K[x_{1},\ldots ,x_{k}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K[x_{1},\ldots ,x_{k}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78d17d2475125ab791247e0a6d107a9a8b203c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.34ex; height:2.843ex;" alt="{\displaystyle K[x_{1},\ldots ,x_{k}]}"></span> which can be interpreted as a variety in <i>k</i>-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="The existence of a common eigenvector is not clear, see https://mathoverflow.net/questions/43298/commuting-matrices-and-the-weak-nullstellensatz (March 2021)">citation needed</span></a></i>]</sup> In algebraic terms, these operators correspond to an <a href="/wiki/Algebra_representation" title="Algebra representation">algebra representation</a> of the polynomial algebra in <i>k</i> variables. </p><p>This is generalized by <a href="/wiki/Lie%27s_theorem" title="Lie's theorem">Lie's theorem</a>, which shows that any representation of a <a href="/wiki/Solvable_Lie_algebra" title="Solvable Lie algebra">solvable Lie algebra</a> is simultaneously upper triangularizable, the case of commuting matrices being the <a href="/wiki/Abelian_Lie_algebra" class="mw-redirect" title="Abelian Lie algebra">abelian Lie algebra</a> case, abelian being a fortiori solvable. </p><p>More generally and precisely, a set of matrices <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_{1},\ldots ,A_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},\ldots ,A_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbe8023d2314d4c8322622f36d0dedc3d93a867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.807ex; height:2.509ex;" alt="{\displaystyle A_{1},\ldots ,A_{k}}"></span> is simultaneously triangularisable if and only 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 p(A_{1},\ldots ,A_{k})[A_{i},A_{j}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(A_{1},\ldots ,A_{k})[A_{i},A_{j}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2d4178296c83210cd64f3e62abadcdbbb73116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:21.399ex; height:3.009ex;" alt="{\displaystyle p(A_{1},\ldots ,A_{k})[A_{i},A_{j}]}"></span> is <a href="/wiki/Nilpotent" title="Nilpotent">nilpotent</a> for all polynomials <i>p</i> in <i>k</i> <i>non</i>-commuting variables, 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_{i},A_{j}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [A_{i},A_{j}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4434b552a3a074fb0813b3c59632e9ce553e30e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.523ex; height:3.009ex;" alt="{\displaystyle [A_{i},A_{j}]}"></span> is the <a href="/wiki/Commutator" title="Commutator">commutator</a>; for commuting <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"></span> the commutator vanishes so this holds. This was proven by Drazin, Dungey, and Gruenberg in 1951;<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> a brief proof is given by Prasolov in 1994.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> One direction is clear: if the matrices are simultaneously triangularisable, 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 [A_{i},A_{j}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [A_{i},A_{j}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4434b552a3a074fb0813b3c59632e9ce553e30e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.523ex; height:3.009ex;" alt="{\displaystyle [A_{i},A_{j}]}"></span> is <i>strictly</i> upper triangularizable (hence nilpotent), which is preserved by multiplication by any <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_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72095229db907e86eb4343cb4736429fcc56507d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.832ex; height:2.509ex;" alt="{\displaystyle A_{k}}"></span> or combination thereof – it will still have 0s on the diagonal in the triangularizing basis. </p> <div class="mw-heading mw-heading2"><h2 id="Algebras_of_triangular_matrices">Algebras of triangular matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=16" title="Edit section: Algebras of triangular matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group_Z4;_Cayley_table;_powers_of_Gray_code_permutation_(small).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Cyclic_group_Z4%3B_Cayley_table%3B_powers_of_Gray_code_permutation_%28small%29.svg/220px-Cyclic_group_Z4%3B_Cayley_table%3B_powers_of_Gray_code_permutation_%28small%29.svg.png" decoding="async" width="220" height="237" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Cyclic_group_Z4%3B_Cayley_table%3B_powers_of_Gray_code_permutation_%28small%29.svg/330px-Cyclic_group_Z4%3B_Cayley_table%3B_powers_of_Gray_code_permutation_%28small%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Cyclic_group_Z4%3B_Cayley_table%3B_powers_of_Gray_code_permutation_%28small%29.svg/440px-Cyclic_group_Z4%3B_Cayley_table%3B_powers_of_Gray_code_permutation_%28small%29.svg.png 2x" data-file-width="478" data-file-height="514" /></a><figcaption><a href="/wiki/Logical_matrix" title="Logical matrix">Binary</a> lower unitriangular <a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz</a> matrices, multiplied using <a href="/wiki/Finite_field" title="Finite field"><b>F</b><sub>2</sub></a> operations. They form the <a href="/wiki/Cayley_table" title="Cayley table">Cayley table</a> of <a href="/wiki/Cyclic_group" title="Cyclic group">Z<sub>4</sub></a> and correspond to <a href="https://en.wikiversity.org/wiki/Gray_code_permutation_powers#4_bit" class="extiw" title="v:Gray code permutation powers">powers of the 4-bit Gray code permutation</a>.</figcaption></figure> <p>Upper triangularity is preserved by many operations: </p> <ul><li>The sum of two upper triangular matrices is upper triangular.</li> <li>The product of two upper triangular matrices is upper triangular.</li> <li>The <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse</a> of an upper triangular matrix, if it exists, is upper triangular.</li> <li>The product of an upper triangular matrix and a scalar is upper triangular.</li></ul> <p>Together these facts mean that the upper triangular matrices form a <a href="/wiki/Subalgebra" title="Subalgebra">subalgebra</a> of the <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of square matrices of a fixed size, where the <a href="/wiki/Lie_bracket" class="mw-redirect" title="Lie bracket">Lie bracket</a> [<i>a</i>, <i>b</i>] given by the <a href="/wiki/Commutator#Ring_theory" title="Commutator">commutator</a> <span class="nowrap"><i>ab − ba</i></span>. The Lie algebra of all upper triangular matrices is a <a href="/wiki/Solvable_Lie_algebra" title="Solvable Lie algebra">solvable Lie algebra</a>. It is often referred to as a <a href="/wiki/Borel_subalgebra" title="Borel subalgebra">Borel subalgebra</a> of the Lie algebra of all square matrices. </p><p>All these results hold if <i>upper triangular</i> is replaced by <i>lower triangular</i> throughout; in particular the lower triangular matrices also form a Lie algebra. However, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a lower triangular matrix can be any matrix; the product of a lower triangular with an upper triangular matrix is not necessarily triangular either. </p><p>The set of unitriangular matrices forms a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>. </p><p>The set of strictly upper (or lower) triangular matrices forms a <a href="/wiki/Nilpotent_Lie_algebra" title="Nilpotent Lie algebra">nilpotent Lie algebra</a>, denoted <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 {\mathfrak {n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">n</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67c5f1431bdaa2f3459db87fb28705821d3c5bbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.872ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {n}}.}"></span> This algebra is the <a href="/wiki/Derived_Lie_algebra" class="mw-redirect" title="Derived Lie algebra">derived Lie algebra</a> 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 {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}"></span>, the Lie algebra of all upper triangular matrices; in symbols, <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 {\mathfrak {n}}=[{\mathfrak {b}},{\mathfrak {b}}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">n</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {n}}=[{\mathfrak {b}},{\mathfrak {b}}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1709b355d7010caf38d5212728a77510886b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.683ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {n}}=[{\mathfrak {b}},{\mathfrak {b}}].}"></span> In addition, <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 {\mathfrak {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">n</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fab30f69b7fab337592fdb8b5384bf004f88c574" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.225ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {n}}}"></span> is the Lie algebra of the Lie group of unitriangular matrices. </p><p>In fact, by <a href="/wiki/Engel%27s_theorem" title="Engel's theorem">Engel's theorem</a>, any finite-dimensional nilpotent Lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices, that is to say, a finite-dimensional nilpotent Lie algebra is simultaneously strictly upper triangularizable. </p><p>Algebras of upper triangular matrices have a natural generalization in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> which yields <a href="/wiki/Nest_algebra" title="Nest algebra">nest algebras</a> on <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Affine_group" title="Affine group">Affine group</a></div> <div class="mw-heading mw-heading3"><h3 id="Borel_subgroups_and_Borel_subalgebras">Borel subgroups and Borel subalgebras</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=17" title="Edit section: Borel subgroups and Borel subalgebras"><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 articles: <a href="/wiki/Borel_subgroup" title="Borel subgroup">Borel subgroup</a> and <a href="/wiki/Borel_subalgebra" title="Borel subalgebra">Borel subalgebra</a></div> <p>The set of invertible triangular matrices of a given kind (upper or lower) forms a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, indeed a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>, which is a subgroup of the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero). </p><p>Over the real numbers, this group is disconnected, having <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 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8226f30650ee4fe4e640c6d2798127e80e9c160d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 2^{n}}"></span> components accordingly as each diagonal entry is positive or negative. The identity component is invertible triangular matrices with positive entries on the diagonal, and the group of all invertible triangular matrices is a <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of this group and the group of <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrices</a> with <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 \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}"></span> on the diagonal, corresponding to the components. </p><p>The <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a <a href="/wiki/Solvable_Lie_algebra" title="Solvable Lie algebra">solvable Lie algebra</a>. These are, respectively, the standard <a href="/wiki/Borel_subgroup" title="Borel subgroup">Borel subgroup</a> <i>B</i> of the Lie group GL<sub><i>n</i></sub> and the standard <a href="/wiki/Borel_subalgebra" title="Borel subalgebra">Borel subalgebra</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 {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}"></span> of the Lie algebra gl<sub><i>n</i></sub>. </p><p>The upper triangular matrices are precisely those that stabilize the <a href="/wiki/Flag_(linear_algebra)" title="Flag (linear algebra)">standard flag</a>. The invertible ones among them form a subgroup of the general linear group, whose conjugate subgroups are those defined as the stabilizer of some (other) complete flag. These subgroups are <a href="/wiki/Borel_subgroup" title="Borel subgroup">Borel subgroups</a>. The group of invertible lower triangular matrices is such a subgroup, since it is the stabilizer of the standard flag associated to the standard basis in reverse order. </p><p>The stabilizer of a partial flag obtained by forgetting some parts of the standard flag can be described as a set of block upper triangular matrices (but its elements are <i>not</i> all triangular matrices). The conjugates of such a group are the subgroups defined as the stabilizer of some partial flag. These subgroups are called parabolic subgroups. </p> <div class="mw-heading mw-heading3"><h3 id="Examples_2">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Triangular_matrix&action=edit&section=18" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The group of 2×2 upper unitriangular matrices is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the <a href="/wiki/Abelian_group" title="Abelian group">additive group</a> of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>; the 3×3 upper unitriangular matrices form the <a href="/wiki/Heisenberg_group" title="Heisenberg group">Heisenberg group</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=Triangular_matrix&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></li> <li><a href="/wiki/QR_decomposition" title="QR decomposition">QR decomposition</a></li> <li><a href="/wiki/Cholesky_decomposition" title="Cholesky decomposition">Cholesky decomposition</a></li> <li><a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg matrix</a></li> <li><a href="/wiki/Tridiagonal_matrix" title="Tridiagonal matrix">Tridiagonal matrix</a></li> <li><a href="/wiki/Invariant_subspace" title="Invariant subspace">Invariant subspace</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=Triangular_matrix&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-axler-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-axler_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-axler_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-axler_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .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="CITEREFAxler1997" class="citation book cs1">Axler, Sheldon Jay (1997). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/54850562"><i>Linear Algebra Done Right</i></a> (2nd ed.). New York: Springer. pp. 86–87, 169. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-22595-1" title="Special:BookSources/0-387-22595-1"><bdi>0-387-22595-1</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/54850562">54850562</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+Done+Right&rft.place=New+York&rft.pages=86-87%2C+169&rft.edition=2nd&rft.pub=Springer&rft.date=1997&rft_id=info%3Aoclcnum%2F54850562&rft.isbn=0-387-22595-1&rft.aulast=Axler&rft.aufirst=Sheldon+Jay&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F54850562&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangular+matrix" class="Z3988"></span></span> </li> <li id="cite_note-bernstein2009-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-bernstein2009_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-bernstein2009_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernstein2009" class="citation book cs1">Bernstein, Dennis S. (2009). <i>Matrix mathematics: theory, facts, and formulas</i> (2 ed.). Princeton, NJ: Princeton University Press. p. 168. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14039-1" title="Special:BookSources/978-0-691-14039-1"><bdi>978-0-691-14039-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+mathematics%3A+theory%2C+facts%2C+and+formulas&rft.place=Princeton%2C+NJ&rft.pages=168&rft.edition=2&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-14039-1&rft.aulast=Bernstein&rft.aufirst=Dennis+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangular+matrix" class="Z3988"></span></span> </li> <li id="cite_note-herstein-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-herstein_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerstein1975" class="citation book cs1">Herstein, I. N. (1975). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/3307396"><i>Topics in Algebra</i></a> (2nd ed.). New York: Wiley. pp. 285–290. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-01090-1" title="Special:BookSources/0-471-01090-1"><bdi>0-471-01090-1</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/3307396">3307396</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+Algebra&rft.place=New+York&rft.pages=285-290&rft.edition=2nd&rft.pub=Wiley&rft.date=1975&rft_id=info%3Aoclcnum%2F3307396&rft.isbn=0-471-01090-1&rft.aulast=Herstein&rft.aufirst=I.+N.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F3307396&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangular+matrix" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDrazinDungeyGruenberg1951" class="citation journal cs1">Drazin, M. P.; Dungey, J. W.; Gruenberg, K. W. (1951). <a rel="nofollow" class="external text" href="http://jlms.oxfordjournals.org/cgi/pdf_extract/s1-26/3/221">"Some Theorems on Commutative Matrices"</a>. <i>Journal of the London Mathematical Society</i>. <b>26</b> (3): 221–228. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fjlms%2Fs1-26.3.221">10.1112/jlms/s1-26.3.221</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+London+Mathematical+Society&rft.atitle=Some+Theorems+on+Commutative+Matrices&rft.volume=26&rft.issue=3&rft.pages=221-228&rft.date=1951&rft_id=info%3Adoi%2F10.1112%2Fjlms%2Fs1-26.3.221&rft.aulast=Drazin&rft.aufirst=M.+P.&rft.au=Dungey%2C+J.+W.&rft.au=Gruenberg%2C+K.+W.&rft_id=http%3A%2F%2Fjlms.oxfordjournals.org%2Fcgi%2Fpdf_extract%2Fs1-26%2F3%2F221&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangular+matrix" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrasolov1994" class="citation book cs1">Prasolov, V. V. (1994). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/30076024"><i>Problems and Theorems in Linear Algebra</i></a>. Simeon Ivanov. Providence, R.I.: American Mathematical Society. pp. 178–179. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821802366" title="Special:BookSources/9780821802366"><bdi>9780821802366</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/30076024">30076024</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Problems+and+Theorems+in+Linear+Algebra&rft.place=Providence%2C+R.I.&rft.pages=178-179&rft.pub=American+Mathematical+Society&rft.date=1994&rft_id=info%3Aoclcnum%2F30076024&rft.isbn=9780821802366&rft.aulast=Prasolov&rft.aufirst=V.+V.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F30076024&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangular+matrix" class="Z3988"></span></span> </li> </ol></div></div> <div 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style="width:1%">Explicitly constrained entries</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternant_matrix" title="Alternant matrix">Alternant</a></li> <li><a href="/wiki/Anti-diagonal_matrix" title="Anti-diagonal matrix">Anti-diagonal</a></li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Anti-Hermitian</a></li> <li><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Anti-symmetric</a></li> <li><a href="/wiki/Arrowhead_matrix" title="Arrowhead matrix">Arrowhead</a></li> <li><a href="/wiki/Band_matrix" title="Band matrix">Band</a></li> <li><a href="/wiki/Bidiagonal_matrix" title="Bidiagonal matrix">Bidiagonal</a></li> <li><a href="/wiki/Bisymmetric_matrix" title="Bisymmetric matrix">Bisymmetric</a></li> <li><a href="/wiki/Block-diagonal_matrix" class="mw-redirect" title="Block-diagonal matrix">Block-diagonal</a></li> <li><a href="/wiki/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Block_tridiagonal_matrix" class="mw-redirect" title="Block tridiagonal matrix">Block tridiagonal</a></li> <li><a href="/wiki/Boolean_matrix" title="Boolean matrix">Boolean</a></li> <li><a href="/wiki/Cauchy_matrix" title="Cauchy matrix">Cauchy</a></li> <li><a href="/wiki/Centrosymmetric_matrix" title="Centrosymmetric matrix">Centrosymmetric</a></li> <li><a href="/wiki/Conference_matrix" title="Conference matrix">Conference</a></li> <li><a href="/wiki/Complex_Hadamard_matrix" title="Complex Hadamard matrix">Complex Hadamard</a></li> <li><a href="/wiki/Copositive_matrix" title="Copositive matrix">Copositive</a></li> <li><a href="/wiki/Diagonally_dominant_matrix" title="Diagonally dominant matrix">Diagonally dominant</a></li> <li><a href="/wiki/Diagonal_matrix" title="Diagonal matrix">Diagonal</a></li> <li><a href="/wiki/DFT_matrix" title="DFT matrix">Discrete Fourier Transform</a></li> <li><a href="/wiki/Elementary_matrix" title="Elementary matrix">Elementary</a></li> <li><a href="/wiki/Equivalent_matrix" class="mw-redirect" title="Equivalent matrix">Equivalent</a></li> <li><a href="/wiki/Frobenius_matrix" title="Frobenius matrix">Frobenius</a></li> <li><a href="/wiki/Generalized_permutation_matrix" title="Generalized permutation matrix">Generalized permutation</a></li> <li><a href="/wiki/Hadamard_matrix" title="Hadamard matrix">Hadamard</a></li> <li><a href="/wiki/Hankel_matrix" title="Hankel matrix">Hankel</a></li> <li><a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a></li> <li><a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg</a></li> <li><a href="/wiki/Hollow_matrix" title="Hollow matrix">Hollow</a></li> <li><a href="/wiki/Integer_matrix" title="Integer matrix">Integer</a></li> <li><a href="/wiki/Logical_matrix" title="Logical matrix">Logical</a></li> <li><a href="/wiki/Matrix_unit" title="Matrix unit">Matrix unit</a></li> <li><a href="/wiki/Metzler_matrix" title="Metzler matrix">Metzler</a></li> <li><a href="/wiki/Moore_matrix" title="Moore matrix">Moore</a></li> <li><a href="/wiki/Nonnegative_matrix" title="Nonnegative matrix">Nonnegative</a></li> <li><a href="/wiki/Pentadiagonal_matrix" class="mw-redirect" title="Pentadiagonal matrix">Pentadiagonal</a></li> <li><a href="/wiki/Permutation_matrix" title="Permutation matrix">Permutation</a></li> <li><a href="/wiki/Persymmetric_matrix" title="Persymmetric matrix">Persymmetric</a></li> <li><a href="/wiki/Polynomial_matrix" title="Polynomial matrix">Polynomial</a></li> <li><a href="/wiki/Quaternionic_matrix" title="Quaternionic matrix">Quaternionic</a></li> <li><a href="/wiki/Signature_matrix" title="Signature matrix">Signature</a></li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian</a></li> <li><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Skew-symmetric</a></li> <li><a href="/wiki/Skyline_matrix" title="Skyline matrix">Skyline</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse</a></li> <li><a href="/wiki/Sylvester_matrix" title="Sylvester matrix">Sylvester</a></li> <li><a href="/wiki/Symmetric_matrix" title="Symmetric matrix">Symmetric</a></li> <li><a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz</a></li> <li><a class="mw-selflink selflink">Triangular</a></li> <li><a href="/wiki/Tridiagonal_matrix" title="Tridiagonal matrix">Tridiagonal</a></li> <li><a href="/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde</a></li> <li><a href="/wiki/Walsh_matrix" title="Walsh matrix">Walsh</a></li> <li><a href="/wiki/Z-matrix_(mathematics)" title="Z-matrix (mathematics)">Z</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constant</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exchange_matrix" title="Exchange matrix">Exchange</a></li> <li><a href="/wiki/Hilbert_matrix" title="Hilbert matrix">Hilbert</a></li> <li><a href="/wiki/Identity_matrix" title="Identity matrix">Identity</a></li> <li><a href="/wiki/Lehmer_matrix" title="Lehmer matrix">Lehmer</a></li> <li><a href="/wiki/Matrix_of_ones" title="Matrix of ones">Of ones</a></li> <li><a href="/wiki/Pascal_matrix" title="Pascal matrix">Pascal</a></li> <li><a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli</a></li> <li><a href="/wiki/Redheffer_matrix" title="Redheffer matrix">Redheffer</a></li> <li><a href="/wiki/Shift_matrix" title="Shift matrix">Shift</a></li> <li><a href="/wiki/Zero_matrix" title="Zero matrix">Zero</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conditions on <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues or eigenvectors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Companion_matrix" title="Companion matrix">Companion</a></li> <li><a href="/wiki/Convergent_matrix" title="Convergent matrix">Convergent</a></li> <li><a href="/wiki/Defective_matrix" title="Defective matrix">Defective</a></li> <li><a href="/wiki/Definite_matrix" title="Definite matrix">Definite</a></li> <li><a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">Diagonalizable</a></li> <li><a href="/wiki/Hurwitz-stable_matrix" title="Hurwitz-stable matrix">Hurwitz-stable</a></li> <li><a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">Positive-definite</a></li> <li><a href="/wiki/Stieltjes_matrix" title="Stieltjes matrix">Stieltjes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satisfying conditions on <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">products</a> or <a href="/wiki/Inverse_of_a_matrix" class="mw-redirect" title="Inverse of a matrix">inverses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Matrix_congruence" title="Matrix congruence">Congruent</a></li> <li><a href="/wiki/Idempotent_matrix" title="Idempotent matrix">Idempotent</a> or <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Projection</a></li> <li><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li> <li><a href="/wiki/Involutory_matrix" title="Involutory matrix">Involutory</a></li> <li><a href="/wiki/Nilpotent_matrix" title="Nilpotent matrix">Nilpotent</a></li> <li><a href="/wiki/Normal_matrix" title="Normal matrix">Normal</a></li> <li><a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">Orthogonal</a></li> <li><a href="/wiki/Unimodular_matrix" title="Unimodular matrix">Unimodular</a></li> <li><a href="/wiki/Unipotent" title="Unipotent">Unipotent</a></li> <li><a href="/wiki/Unitary_matrix" title="Unitary matrix">Unitary</a></li> <li><a href="/wiki/Totally_unimodular_matrix" class="mw-redirect" title="Totally unimodular matrix">Totally unimodular</a></li> <li><a href="/wiki/Weighing_matrix" title="Weighing matrix">Weighing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With specific applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjugate_matrix" title="Adjugate matrix">Adjugate</a></li> <li><a href="/wiki/Alternating_sign_matrix" title="Alternating sign matrix">Alternating sign</a></li> <li><a href="/wiki/Augmented_matrix" title="Augmented matrix">Augmented</a></li> <li><a href="/wiki/B%C3%A9zout_matrix" title="Bézout matrix">Bézout</a></li> <li><a href="/wiki/Carleman_matrix" title="Carleman matrix">Carleman</a></li> <li><a href="/wiki/Cartan_matrix" title="Cartan matrix">Cartan</a></li> <li><a href="/wiki/Circulant_matrix" title="Circulant matrix">Circulant</a></li> <li><a href="/wiki/Cofactor_matrix" class="mw-redirect" title="Cofactor matrix">Cofactor</a></li> <li><a href="/wiki/Commutation_matrix" title="Commutation matrix">Commutation</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion</a></li> <li><a href="/wiki/Coxeter_matrix" class="mw-redirect" title="Coxeter matrix">Coxeter</a></li> <li><a href="/wiki/Distance_matrix" title="Distance matrix">Distance</a></li> <li><a href="/wiki/Duplication_and_elimination_matrices" title="Duplication and elimination matrices">Duplication and elimination</a></li> <li><a href="/wiki/Euclidean_distance_matrix" title="Euclidean distance matrix">Euclidean distance</a></li> <li><a href="/wiki/Fundamental_matrix_(linear_differential_equation)" title="Fundamental matrix (linear differential equation)">Fundamental (linear differential equation)</a></li> <li><a href="/wiki/Generator_matrix" title="Generator matrix">Generator</a></li> <li><a href="/wiki/Gram_matrix" title="Gram matrix">Gram</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li> <li><a href="/wiki/Householder_transformation" title="Householder transformation">Householder</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Moment_matrix" title="Moment matrix">Moment</a></li> <li><a href="/wiki/Payoff_matrix" class="mw-redirect" title="Payoff matrix">Payoff</a></li> <li><a href="/wiki/Pick_matrix" class="mw-redirect" title="Pick matrix">Pick</a></li> <li><a href="/wiki/Random_matrix" title="Random matrix">Random</a></li> <li><a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation</a></li> <li><a href="/wiki/Routh%E2%80%93Hurwitz_matrix" title="Routh–Hurwitz matrix">Routh-Hurwitz</a></li> <li><a href="/wiki/Seifert_matrix" class="mw-redirect" title="Seifert matrix">Seifert</a></li> <li><a href="/wiki/Shear_matrix" class="mw-redirect" title="Shear matrix">Shear</a></li> <li><a href="/wiki/Similarity_matrix" class="mw-redirect" title="Similarity matrix">Similarity</a></li> <li><a href="/wiki/Symplectic_matrix" title="Symplectic matrix">Symplectic</a></li> <li><a href="/wiki/Totally_positive_matrix" title="Totally positive matrix">Totally positive</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/wiki/Statistics" title="Statistics">statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centering_matrix" title="Centering matrix">Centering</a></li> <li><a href="/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">Correlation</a></li> <li><a href="/wiki/Covariance_matrix" title="Covariance matrix">Covariance</a></li> <li><a href="/wiki/Design_matrix" title="Design matrix">Design</a></li> <li><a href="/wiki/Doubly_stochastic_matrix" title="Doubly stochastic matrix">Doubly stochastic</a></li> <li><a href="/wiki/Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information</a></li> <li><a href="/wiki/Projection_matrix" title="Projection matrix">Hat</a></li> <li><a href="/wiki/Precision_(statistics)" title="Precision (statistics)">Precision</a></li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Stochastic</a></li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Transition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjacency_matrix" title="Adjacency matrix">Adjacency</a></li> <li><a href="/wiki/Biadjacency_matrix" class="mw-redirect" title="Biadjacency matrix">Biadjacency</a></li> <li><a href="/wiki/Degree_matrix" title="Degree matrix">Degree</a></li> <li><a href="/wiki/Edmonds_matrix" title="Edmonds matrix">Edmonds</a></li> <li><a href="/wiki/Incidence_matrix" title="Incidence matrix">Incidence</a></li> <li><a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian</a></li> <li><a href="/wiki/Seidel_adjacency_matrix" title="Seidel adjacency matrix">Seidel adjacency</a></li> <li><a href="/wiki/Tutte_matrix" title="Tutte matrix">Tutte</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in science and engineering</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density</a></li> <li><a href="/wiki/Fundamental_matrix_(computer_vision)" title="Fundamental matrix (computer vision)">Fundamental (computer vision)</a></li> <li><a href="/wiki/Fuzzy_associative_matrix" title="Fuzzy associative matrix">Fuzzy associative</a></li> <li><a href="/wiki/Gamma_matrices" title="Gamma matrices">Gamma</a></li> <li><a href="/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann</a></li> <li><a href="/wiki/Hamiltonian_matrix" title="Hamiltonian matrix">Hamiltonian</a></li> <li><a href="/wiki/Irregular_matrix" title="Irregular matrix">Irregular</a></li> <li><a href="/wiki/Overlap_matrix" class="mw-redirect" title="Overlap matrix">Overlap</a></li> <li><a href="/wiki/S-matrix" title="S-matrix">S</a></li> <li><a href="/wiki/State-transition_matrix" title="State-transition matrix">State transition</a></li> <li><a href="/wiki/Substitution_matrix" title="Substitution matrix">Substitution</a></li> <li><a href="/wiki/Z-matrix_(chemistry)" title="Z-matrix (chemistry)">Z (chemistry)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related terms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Matrix_exponential" title="Matrix exponential">Matrix exponential</a></li> <li><a href="/wiki/Matrix_representation_of_conic_sections" title="Matrix representation of conic sections">Matrix representation of conic sections</a></li> <li><a href="/wiki/Perfect_matrix" title="Perfect matrix">Perfect matrix</a></li> <li><a href="/wiki/Pseudoinverse" class="mw-redirect" title="Pseudoinverse">Pseudoinverse</a></li> <li><a href="/wiki/Row_echelon_form" title="Row echelon form">Row echelon form</a></li> <li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" 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Triangular matrix - Wikipedia