Orthogonal matrix - Wikipedia

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id="toc-Lower_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Higher dimensions</span> </div> </a> <ul id="toc-Higher_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primitives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primitives"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Primitives</span> </div> </a> <ul id="toc-Primitives-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">4</span> <span>Properties</span> </div> </a> <button 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vector-toc-level-2"> <a class="vector-toc-link" href="#Canonical_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Canonical form</span> </div> </a> <ul id="toc-Canonical_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lie_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Lie algebra</span> </div> </a> <ul id="toc-Lie_algebra-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Numerical_linear_algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Numerical_linear_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Numerical linear algebra</span> </div> </a> <button aria-controls="toc-Numerical_linear_algebra-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 Numerical linear algebra subsection</span> </button> <ul id="toc-Numerical_linear_algebra-sublist" class="vector-toc-list"> <li id="toc-Benefits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Benefits"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Benefits</span> </div> </a> <ul id="toc-Benefits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decompositions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decompositions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Decompositions</span> </div> </a> <ul id="toc-Decompositions-sublist" class="vector-toc-list"> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2.1</span> <span>Examples</span> </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Randomization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Randomization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Randomization</span> </div> </a> <ul id="toc-Randomization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nearest_orthogonal_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nearest_orthogonal_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Nearest orthogonal matrix</span> </div> </a> <ul id="toc-Nearest_orthogonal_matrix-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Spin_and_pin" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spin_and_pin"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Spin and pin</span> </div> </a> <ul id="toc-Spin_and_pin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rectangular_matrices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rectangular_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Rectangular matrices</span> </div> </a> <ul id="toc-Rectangular_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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">Orthogonal 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 31 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-31" 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">31 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%AA%D8%B9%D8%A7%D9%85%D8%AF%D8%A9" title="مصفوفة متعامدة – Arabic" lang="ar" hreflang="ar" data-title="مصفوفة متعامدة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ti%CC%8Dt-kau_h%C3%A2ng-lia%CC%8Dt" title="Ti̍t-kau hâng-lia̍t – Minnan" lang="nan" hreflang="nan" data-title="Ti̍t-kau hâng-lia̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D1%80%D1%82%D0%B0%D0%B3%D0%B0%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0" title="Артаганальная матрыца – Belarusian" lang="be" hreflang="be" data-title="Артаганальная матрыца" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matriu_ortogonal" title="Matriu ortogonal – Catalan" lang="ca" hreflang="ca" data-title="Matriu ortogonal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BB%C4%95_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Ортогоналлĕ матрица – Chuvash" lang="cv" hreflang="cv" data-title="Ортогоналлĕ матрица" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Ortogon%C3%A1ln%C3%AD_matice" title="Ortogonální matice – Czech" lang="cs" hreflang="cs" data-title="Ortogonální matice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ortogonal_matrix" title="Ortogonal matrix – Danish" lang="da" hreflang="da" data-title="Ortogonal matrix" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Orthogonale_Matrix" title="Orthogonale Matrix – German" lang="de" hreflang="de" data-title="Orthogonale Matrix" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CF%81%CE%B8%CE%BF%CE%B3%CF%8E%CE%BD%CE%B9%CE%BF%CF%82_%CF%80%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82" title="Ορθογώνιος πίνακας – Greek" lang="el" hreflang="el" data-title="Ορθογώνιος 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_ortogonal" title="Matriz ortogonal – Spanish" lang="es" hreflang="es" data-title="Matriz ortogonal" 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/Orta_matrico" title="Orta matrico – Esperanto" lang="eo" hreflang="eo" data-title="Orta 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_ortogonal" title="Matrize ortogonal – Basque" lang="eu" hreflang="eu" data-title="Matrize ortogonal" 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%AA%D8%B9%D8%A7%D9%85%D8%AF" title="ماتریس متعامد – Persian" lang="fa" hreflang="fa" data-title="ماتریس متعامد" 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_orthogonale" title="Matrice orthogonale – French" lang="fr" hreflang="fr" data-title="Matrice orthogonale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%81%EA%B5%90%ED%96%89%EB%A0%AC" title="직교행렬 – Korean" lang="ko" hreflang="ko" data-title="직교행렬" 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_ortogonal" title="Matriks ortogonal – Indonesian" lang="id" hreflang="id" data-title="Matriks ortogonal" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matrice_ortogonale" title="Matrice ortogonale – Italian" lang="it" hreflang="it" data-title="Matrice ortogonale" 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%90%D7%95%D7%A8%D7%AA%D7%95%D7%92%D7%95%D7%A0%D7%9C%D7%99%D7%AA" title="מטריצה אורתוגונלית – Hebrew" lang="he" hreflang="he" data-title="מטריצה אורתוגונלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ortogon%C3%A1lis_m%C3%A1trix" title="Ortogonális mátrix – Hungarian" lang="hu" hreflang="hu" data-title="Ortogonális mátrix" 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/Orthogonale_matrix" title="Orthogonale matrix – Dutch" lang="nl" hreflang="nl" data-title="Orthogonale matrix" 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/%E7%9B%B4%E4%BA%A4%E8%A1%8C%E5%88%97" title="直交行列 – Japanese" lang="ja" hreflang="ja" data-title="直交行列" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Macierz_ortogonalna" title="Macierz ortogonalna – Polish" lang="pl" hreflang="pl" data-title="Macierz ortogonalna" 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_ortogonal" title="Matriz ortogonal – Portuguese" lang="pt" hreflang="pt" data-title="Matriz ortogonal" 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%9E%D1%80%D1%82%D0%BE%D0%B3%D0%BE%D0%BD%D0%B0%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="Ортогональная матрица – Russian" lang="ru" hreflang="ru" data-title="Ортогональная 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/Ortogonalna_matrika" title="Ortogonalna matrika – Slovenian" lang="sl" hreflang="sl" data-title="Ortogonalna 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/Ortogonaalinen_matriisi" title="Ortogonaalinen matriisi – Finnish" lang="fi" hreflang="fi" data-title="Ortogonaalinen matriisi" 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/Ortogonalmatris" title="Ortogonalmatris – Swedish" lang="sv" hreflang="sv" data-title="Ortogonalmatris" 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%9A%E0%AF%86%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AF%81_%E0%AE%85%E0%AE%A3%E0%AE%BF" title="செங்குத்து அணி – Tamil" lang="ta" hreflang="ta" data-title="செங்குத்து அணி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Real square matrix whose columns and rows are orthogonal unit vectors</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For matrices with orthogonality over the <a href="/wiki/Complex_number" title="Complex number">complex number</a> field, see <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">May 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, an <b>orthogonal matrix</b>, or <b>orthonormal matrix</b>, is a real <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> whose columns and rows are <a href="/wiki/Orthonormality" title="Orthonormality">orthonormal</a> <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a>. </p><p>One way to express this is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> <mo>=</mo> <mi>Q</mi> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bace68ad0f492ab10e5f60fdb496159d49daf2ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.206ex; height:3.009ex;" alt="{\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}"></span> where <span class="texhtml"><i>Q</i><sup>T</sup></span> is the <a href="/wiki/Transpose" title="Transpose">transpose</a> of <span class="texhtml mvar" style="font-style:italic;">Q</span> and <span class="texhtml mvar" style="font-style:italic;">I</span> is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>. </p><p>This leads to the equivalent characterization: a matrix <span class="texhtml mvar" style="font-style:italic;">Q</span> is orthogonal if its transpose is equal to its <a href="/wiki/Invertible_matrix" title="Invertible matrix">inverse</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{\mathrm {T} }=Q^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{\mathrm {T} }=Q^{-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124eea429cef65f94d69c98f933c2c174ebe3aaf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.174ex; height:3.009ex;" alt="{\displaystyle Q^{\mathrm {T} }=Q^{-1},}"></span> where <span class="texhtml"><i>Q</i><sup>−1</sup></span> is the inverse of <span class="texhtml mvar" style="font-style:italic;">Q</span>. </p><p>An orthogonal matrix <span class="texhtml mvar" style="font-style:italic;">Q</span> is necessarily invertible (with inverse <span class="texhtml"><i>Q</i><sup>−1</sup> = <i>Q</i><sup>T</sup></span>), <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary</a> (<span class="texhtml"><i>Q</i><sup>−1</sup> = <i>Q</i><sup>∗</sup></span>), where <span class="texhtml"><i>Q</i><sup>∗</sup></span> is the <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Hermitian adjoint</a> (<a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>) of <span class="texhtml mvar" style="font-style:italic;">Q</span>, and therefore <a href="/wiki/Normal_matrix" title="Normal matrix">normal</a> (<span class="texhtml"><i>Q</i><sup>∗</sup><i>Q</i> = <i>QQ</i><sup>∗</sup></span>) over the <a href="/wiki/Real_number" title="Real number">real numbers</a>. The <a href="/wiki/Determinant" title="Determinant">determinant</a> of any orthogonal matrix is either +1 or −1. As a <a href="/wiki/Linear_map" title="Linear map">linear transformation</a>, an orthogonal matrix preserves the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> of vectors, and therefore acts as an <a href="/wiki/Isometry" title="Isometry">isometry</a> of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, such as a <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a>, <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> or <a href="/wiki/Improper_rotation" title="Improper rotation">rotoreflection</a>. In other words, it is a <a href="/wiki/Unitary_transformation" title="Unitary transformation">unitary transformation</a>. </p><p>The set of <span class="texhtml"><i>n</i> × <i>n</i></span> orthogonal matrices, under multiplication, forms the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <span class="texhtml">O(<i>n</i>)</span>, known as the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>. The <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> <span class="texhtml">SO(<i>n</i>)</span> consisting of orthogonal matrices with determinant +1 is called the <a href="/wiki/Orthogonal_group#special_orthogonal_group" title="Orthogonal group">special orthogonal group</a>, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Matrix_multiplication_transpose.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Matrix_multiplication_transpose.svg/275px-Matrix_multiplication_transpose.svg.png" decoding="async" width="275" height="376" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Matrix_multiplication_transpose.svg/413px-Matrix_multiplication_transpose.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Matrix_multiplication_transpose.svg/550px-Matrix_multiplication_transpose.svg.png 2x" data-file-width="458" data-file-height="627" /></a><figcaption>Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA<sup>T</sup> will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of A.</figcaption></figure> <p>An orthogonal matrix is the real specialization of a unitary matrix, and thus always a <a href="/wiki/Normal_matrix" title="Normal matrix">normal matrix</a>. Although we consider only real matrices here, the definition can be used for matrices with entries from any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. However, orthogonal matrices arise naturally from <a href="/wiki/Dot_product" title="Dot product">dot products</a>, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> so, for vectors <span class="texhtml"><b>u</b></span> and <span class="texhtml"><b>v</b></span> in an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional real <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {u} }\cdot {\mathbf {v} }=\left(Q{\mathbf {u} }\right)\cdot \left(Q{\mathbf {v} }\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {u} }\cdot {\mathbf {v} }=\left(Q{\mathbf {u} }\right)\cdot \left(Q{\mathbf {v} }\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3522dee4e02d0da3bfe6e68991795b9736481bf4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.544ex; height:2.843ex;" alt="{\displaystyle {\mathbf {u} }\cdot {\mathbf {v} }=\left(Q{\mathbf {u} }\right)\cdot \left(Q{\mathbf {v} }\right)}"></span> where <span class="texhtml mvar" style="font-style:italic;">Q</span> is an orthogonal matrix. To see the inner product connection, consider a vector <span class="texhtml"><b>v</b></span> in an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional real <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. Written with respect to an orthonormal basis, the squared length of <span class="texhtml"><b>v</b></span> is <span class="texhtml"><b>v</b><sup>T</sup><b>v</b></span>. If a linear transformation, in matrix form <span class="texhtml"><i>Q</i><b>v</b></span>, preserves vector lengths, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }=(Q{\mathbf {v} })^{\mathrm {T} }(Q{\mathbf {v} })={\mathbf {v} }^{\mathrm {T} }Q^{\mathrm {T} }Q{\mathbf {v} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }=(Q{\mathbf {v} })^{\mathrm {T} }(Q{\mathbf {v} })={\mathbf {v} }^{\mathrm {T} }Q^{\mathrm {T} }Q{\mathbf {v} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1b5b19f6c33702f3ef9c72204b10682708bb9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.957ex; height:3.176ex;" alt="{\displaystyle {\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }=(Q{\mathbf {v} })^{\mathrm {T} }(Q{\mathbf {v} })={\mathbf {v} }^{\mathrm {T} }Q^{\mathrm {T} }Q{\mathbf {v} }.}"></span> </p><p>Thus <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">finite-dimensional</a> linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent. </p><p>Orthogonal matrices are important for a number of reasons, both theoretical and practical. The <span class="texhtml"><i>n</i> × <i>n</i></span> orthogonal matrices form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> under matrix multiplication, the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> denoted by <span class="texhtml">O(<i>n</i>)</span>, which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the <a href="/wiki/Point_group" title="Point group">point group</a> of a molecule is a subgroup of O(3). Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algorithms in numerical linear algebra, such as <a href="/wiki/QR_decomposition" title="QR decomposition"><span class="texhtml mvar" style="font-style:italic;">QR</span> decomposition</a>. As another example, with appropriate normalization the <a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">discrete cosine transform</a> (used in <a href="/wiki/MP3" title="MP3">MP3</a> compression) is represented by an orthogonal matrix. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Below are a few examples of small orthogonal matrices and possible interpretations. </p> <ul><li><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&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>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e5393554ca6242bd3e1e35e259e3c0b1592d2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.854ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}}"></span>    (identity transformation)</li> <li><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}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9516427f6ddf2d696e90b15be78c523f4b93c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.646ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}"></span>    (rotation about the origin)</li> <li><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&-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>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0\\0&-1\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d84c06e539c28fb7b67837cca237262b2a7ed889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.662ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0\\0&-1\\\end{bmatrix}}}"></span>    (reflection across <i>x</i>-axis)</li> <li><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}0&0&0&1\\0&0&1&0\\1&0&0&0\\0&1&0&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&0&0&1\\0&0&1&0\\1&0&0&0\\0&1&0&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b9b83dc0ea8ff62b06e2badae2f037b5a2d0d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:15.47ex; height:12.509ex;" alt="{\displaystyle {\begin{bmatrix}0&0&0&1\\0&0&1&0\\1&0&0&0\\0&1&0&0\end{bmatrix}}}"></span>    (permutation of coordinate axes)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Elementary_constructions">Elementary constructions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=3" title="Edit section: Elementary constructions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Lower_dimensions">Lower dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=4" title="Edit section: Lower dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest orthogonal matrices are the <span class="nowrap">1 × 1</span> matrices [1] and [−1], which we can interpret as the identity and a reflection of the real line across the origin. </p><p>The <span class="nowrap">2 × 2</span> matrices have the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}p&t\\q&u\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi>u</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}p&t\\q&u\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2be46b6c6bb30da41571349cc3ac14c10e6e43" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.675ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}p&t\\q&u\end{bmatrix}},}"></span> which orthogonality demands satisfy the three equations <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}1&=p^{2}+t^{2},\\1&=q^{2}+u^{2},\\0&=pq+tu.\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> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>p</mi> <mi>q</mi> <mo>+</mo> <mi>t</mi> <mi>u</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}1&=p^{2}+t^{2},\\1&=q^{2}+u^{2},\\0&=pq+tu.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881f480a77250727cb659a92e4c2e94ec77d5c4d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.017ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}1&=p^{2}+t^{2},\\1&=q^{2}+u^{2},\\0&=pq+tu.\end{aligned}}}"></span> </p><p>In consideration of the first equation, without loss of generality let <span class="texhtml"><i>p</i> = cos <i>θ</i></span>, <span class="texhtml"><i>q</i> = sin <i>θ</i></span>; then either <span class="texhtml"><i>t</i> = −<i>q</i></span>, <span class="texhtml"><i>u</i> = <i>p</i></span> or <span class="texhtml"><i>t</i> = <i>q</i></span>, <span class="texhtml"><i>u</i> = −<i>p</i></span>. We can interpret the first case as a rotation by <span class="texhtml mvar" style="font-style:italic;">θ</span> (where <span class="texhtml"><i>θ</i> = 0</span> is the identity), and the second as a reflection across a line at an angle of <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>θ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\text{ (rotation), }}\qquad {\begin{bmatrix}\cos \theta &\sin \theta \\\sin \theta &-\cos \theta \\\end{bmatrix}}{\text{ (reflection)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> (rotation), </mtext> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> (reflection)</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\text{ (rotation), }}\qquad {\begin{bmatrix}\cos \theta &\sin \theta \\\sin \theta &-\cos \theta \\\end{bmatrix}}{\text{ (reflection)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de88a0eaf30551b9e7a141e377e07bf05acdc8f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.722ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\text{ (rotation), }}\qquad {\begin{bmatrix}\cos \theta &\sin \theta \\\sin \theta &-\cos \theta \\\end{bmatrix}}{\text{ (reflection)}}}"></span> </p><p>The special case of the reflection matrix with <span class="texhtml"><i>θ</i> = 90°</span> generates a reflection about the line at 45° given by <span class="texhtml"><i>y</i> = <i>x</i></span> and therefore exchanges <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>; it is a <a href="/wiki/Permutation_matrix" title="Permutation matrix">permutation matrix</a>, with a single 1 in each column and row (and otherwise 0): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3891f816e1034f79d9c21b63596bad0308f48ca5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.501ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}.}"></span> </p><p>The identity is also a permutation matrix. </p><p>A reflection is <a href="/wiki/Involutory_matrix" title="Involutory matrix">its own inverse</a>, which implies that a reflection matrix is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrix</a>, and the product of two reflection matrices is also a rotation matrix. </p> <div class="mw-heading mw-heading3"><h3 id="Higher_dimensions">Higher dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=5" title="Edit section: Higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for <span class="nowrap">3 × 3</span> matrices and larger the non-rotational matrices can be more complicated than reflections. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}{\text{ and }}{\begin{bmatrix}0&-1&0\\1&0&0\\0&0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}{\text{ and }}{\begin{bmatrix}0&-1&0\\1&0&0\\0&0&-1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d7177a4a95ca41d1a61e54b7c1bfe24d171b3d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:37.919ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}{\text{ and }}{\begin{bmatrix}0&-1&0\\1&0&0\\0&0&-1\end{bmatrix}}}"></span> </p><p>represent an <i><a href="/wiki/Inversion_in_a_point" class="mw-redirect" title="Inversion in a point">inversion</a></i> through the origin and a <i><a href="/wiki/Improper_rotation" title="Improper rotation">rotoinversion</a></i>, respectively, about the <span class="texhtml">z</span>-axis. </p><p>Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a <span class="nowrap">3 × 3</span> rotation matrix in terms of an <a href="/wiki/Axis_and_angle" class="mw-redirect" title="Axis and angle">axis and angle</a>, but this only works in three dimensions. Above three dimensions two or more angles are needed, each associated with a <a href="/wiki/Plane_of_rotation" title="Plane of rotation">plane of rotation</a>. </p><p>However, we have elementary building blocks for permutations, reflections, and rotations that apply in general. </p> <div class="mw-heading mw-heading3"><h3 id="Primitives">Primitives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=6" title="Edit section: Primitives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any <span class="texhtml"><i>n</i> × <i>n</i></span> permutation matrix can be constructed as a product of no more than <span class="texhtml"><i>n</i> − 1</span> transpositions. </p><p>A <a href="/wiki/Householder_reflection" class="mw-redirect" title="Householder reflection">Householder reflection</a> is constructed from a non-null vector <span class="texhtml"><b>v</b></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=I-2{\frac {{\mathbf {v} }{\mathbf {v} }^{\mathrm {T} }}{{\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>I</mi> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=I-2{\frac {{\mathbf {v} }{\mathbf {v} }^{\mathrm {T} }}{{\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9118ea4d3718e1900832ac21fb89ddcae4331b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.835ex; height:6.009ex;" alt="{\displaystyle Q=I-2{\frac {{\mathbf {v} }{\mathbf {v} }^{\mathrm {T} }}{{\mathbf {v} }^{\mathrm {T} }{\mathbf {v} }}}.}"></span> </p><p>Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of <span class="texhtml"><b>v</b></span>. This is a reflection in the hyperplane perpendicular to <span class="texhtml"><b>v</b></span> (negating any vector component parallel to <span class="texhtml"><b>v</b></span>). If <span class="texhtml"><b>v</b></span> is a unit vector, then <span class="texhtml"><i>Q</i> = <i>I</i> − 2<b>vv</b><sup>T</sup></span> suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size <span class="nowrap"><i>n</i> × <i>n</i></span> can be constructed as a product of at most <span class="texhtml mvar" style="font-style:italic;">n</span> such reflections. </p><p>A <a href="/wiki/Givens_rotation" title="Givens rotation">Givens rotation</a> acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size <span class="texhtml"><i>n</i> × <i>n</i></span> can be constructed as a product of at most <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i>(<i>n</i> − 1)</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> such rotations. In the case of <span class="nowrap">3 × 3</span> matrices, three such rotations suffice; and by fixing the sequence we can thus describe all <span class="nowrap">3 × 3</span> rotation matrices (though not uniquely) in terms of the three angles used, often called <a href="/wiki/Euler_angles" title="Euler angles">Euler angles</a>. </p><p>A <a href="/wiki/Jacobi_rotation" title="Jacobi rotation">Jacobi rotation</a> has the same form as a Givens rotation, but is used to zero both off-diagonal entries of a <span class="nowrap">2 × 2</span> symmetric submatrix. </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=Orthogonal_matrix&action=edit&section=7" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Matrix_properties">Matrix properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=8" title="Edit section: Matrix properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A real square matrix is orthogonal <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> its columns form an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> of the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> with the ordinary Euclidean <a href="/wiki/Dot_product" title="Dot product">dot product</a>, which is the case if and only if its rows form an orthonormal basis of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>. It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy <span class="texhtml"><i>M</i><sup>T</sup><i>M</i> = <i>D</i></span>, with <span class="texhtml mvar" style="font-style:italic;">D</span> a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a>. </p><p>The <a href="/wiki/Determinant" title="Determinant">determinant</a> of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=\det(I)=\det \left(Q^{\mathrm {T} }Q\right)=\det \left(Q^{\mathrm {T} }\right)\det(Q)={\bigl (}\det(Q){\bigr )}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=\det(I)=\det \left(Q^{\mathrm {T} }Q\right)=\det \left(Q^{\mathrm {T} }\right)\det(Q)={\bigl (}\det(Q){\bigr )}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821f2b797f530f042297de63ec66c018ff4bc410" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:57.972ex; height:3.676ex;" alt="{\displaystyle 1=\det(I)=\det \left(Q^{\mathrm {T} }Q\right)=\det \left(Q^{\mathrm {T} }\right)\det(Q)={\bigl (}\det(Q){\bigr )}^{2}.}"></span> </p><p>The converse is not true; having a determinant of ±1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}2&0\\0&{\frac {1}{2}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}2&0\\0&{\frac {1}{2}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e7ca0d151e8c172a5988b902f3e509718e4740" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.605ex; height:7.509ex;" alt="{\displaystyle {\begin{bmatrix}2&0\\0&{\frac {1}{2}}\end{bmatrix}}}"></span> </p><p>With permutation matrices the determinant matches the <a href="/wiki/Even_and_odd_permutations" class="mw-redirect" title="Even and odd permutations">signature</a>, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows. </p><p>Stronger than the determinant restriction is the fact that an orthogonal matrix can always be <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalized</a> over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> to exhibit a full set of <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>, all of which must have (complex) <a href="/wiki/Absolute_value" title="Absolute value">modulus</a> 1. </p> <div class="mw-heading mw-heading3"><h3 id="Group_properties">Group properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=9" title="Edit section: Group properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all <span class="texhtml"><i>n</i> × <i>n</i></span> orthogonal matrices satisfies all the axioms of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>. It is a <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Lie_group" title="Lie group">Lie group</a> of dimension <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i>(<i>n</i> − 1)</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, called the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> and denoted by <span class="texhtml">O(<i>n</i>)</span>. </p><p>The orthogonal matrices whose determinant is +1 form a <a href="/wiki/Connected_space" title="Connected space">path-connected</a> <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of <span class="texhtml">O(<i>n</i>)</span> of <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">index</a> 2, the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a> <span class="texhtml">SO(<i>n</i>)</span> of rotations. The <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> <span class="texhtml">O(<i>n</i>)/SO(<i>n</i>)</span> is isomorphic to <span class="texhtml">O(1)</span>, with the projection map choosing [+1] or [−1] according to the determinant. Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a <a href="/wiki/Coset" title="Coset">coset</a>; it is also (separately) connected. Thus each orthogonal group falls into two pieces; and because the projection map <a href="/wiki/Exact_sequence" title="Exact sequence">splits</a>, <span class="texhtml">O(<i>n</i>)</span> is a <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of <span class="texhtml">SO(<i>n</i>)</span> by <span class="texhtml">O(1)</span>. In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with <span class="nowrap">2 × 2</span> matrices. If <span class="texhtml mvar" style="font-style:italic;">n</span> is odd, then the semidirect product is in fact a <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a>, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns. This follows from the property of determinants that negating a column negates the determinant, and thus negating an odd (but not even) number of columns negates the determinant. </p><p>Now consider <span class="texhtml">(<i>n</i> + 1) × (<i>n</i> + 1)</span> orthogonal matrices with bottom right entry equal to 1. The remainder of the last column (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of the matrix is an <span class="texhtml"><i>n</i> × <i>n</i></span> orthogonal matrix; thus <span class="texhtml">O(<i>n</i>)</span> is a subgroup of <span class="texhtml">O(<i>n</i> + 1)</span> (and of all higher groups). </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}&&&0\\&\mathrm {O} (n)&&\vdots \\&&&0\\0&\cdots &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 /> <mtd /> <mtd /> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd /> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </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}&&&0\\&\mathrm {O} (n)&&\vdots \\&&&0\\0&\cdots &0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b56f5dee34b0b194edbefc40aff7fafff46df5f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:19.319ex; height:13.843ex;" alt="{\displaystyle {\begin{bmatrix}&&&0\\&\mathrm {O} (n)&&\vdots \\&&&0\\0&\cdots &0&1\end{bmatrix}}}"></span> </p><p>Since an elementary reflection in the form of a <a href="/wiki/Householder_matrix" class="mw-redirect" title="Householder matrix">Householder matrix</a> can reduce any orthogonal matrix to this constrained form, a series of such reflections can bring any orthogonal matrix to the identity; thus an orthogonal group is a <a href="/wiki/Reflection_group" title="Reflection group">reflection group</a>. The last column can be fixed to any unit vector, and each choice gives a different copy of <span class="texhtml">O(<i>n</i>)</span> in <span class="texhtml">O(<i>n</i> + 1)</span>; in this way <span class="texhtml">O(<i>n</i> + 1)</span> is a <a href="/wiki/Fiber_bundle" title="Fiber bundle">bundle</a> over the unit sphere <span class="texhtml"><i>S</i><sup><i>n</i></sup></span> with fiber <span class="texhtml">O(<i>n</i>)</span>. </p><p>Similarly, <span class="texhtml">SO(<i>n</i>)</span> is a subgroup of <span class="texhtml">SO(<i>n</i> + 1)</span>; and any special orthogonal matrix can be generated by <a href="/wiki/Givens_rotation" title="Givens rotation">Givens plane rotations</a> using an analogous procedure. The bundle structure persists: <span class="texhtml">SO(<i>n</i>) ↪ SO(<i>n</i> + 1) → <i>S</i><sup><i>n</i></sup></span>. A single rotation can produce a zero in the first row of the last column, and series of <span class="texhtml"><i>n</i> − 1</span> rotations will zero all but the last row of the last column of an <span class="texhtml"><i>n</i> × <i>n</i></span> rotation matrix. Since the planes are fixed, each rotation has only one degree of freedom, its angle. By induction, <span class="texhtml">SO(<i>n</i>)</span> therefore has <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)+(n-2)+\cdots +1={\frac {n(n-1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)+(n-2)+\cdots +1={\frac {n(n-1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17bd3d5d70c0edc6bd56cd4719f11eb8c3fb1772" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.357ex; height:5.676ex;" alt="{\displaystyle (n-1)+(n-2)+\cdots +1={\frac {n(n-1)}{2}}}"></span> degrees of freedom, and so does <span class="texhtml">O(<i>n</i>)</span>. </p><p>Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order <a href="/wiki/Factorial" title="Factorial"><span class="texhtml"><i>n</i>!</span></a> <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> <span class="texhtml">S<sub><i>n</i></sub></span>. By the same kind of argument, <span class="texhtml">S<sub><i>n</i></sub></span> is a subgroup of <span class="texhtml">S<sub><i>n</i> + 1</sub></span>. The even permutations produce the subgroup of permutation matrices of determinant +1, the order <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i>!</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Canonical_form">Canonical form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=10" title="Edit section: Canonical form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if <span class="texhtml mvar" style="font-style:italic;">Q</span> is special orthogonal then one can always find an orthogonal matrix <span class="texhtml mvar" style="font-style:italic;">P</span>, a (rotational) <a href="/wiki/Change_of_basis" title="Change of basis">change of basis</a>, that brings <span class="texhtml mvar" style="font-style:italic;">Q</span> into block diagonal form: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{bmatrix}}\ (n{\text{ even}}),\ P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&&\\&\ddots &&\\&&R_{k}&\\&&&1\end{bmatrix}}\ (n{\text{ odd}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even</mtext> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd /> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> odd</mtext> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{bmatrix}}\ (n{\text{ even}}),\ P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&&\\&\ddots &&\\&&R_{k}&\\&&&1\end{bmatrix}}\ (n{\text{ odd}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d9ea2f8d41b5d11b059520ef0f45c90605c50e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:76.948ex; height:14.176ex;" alt="{\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{bmatrix}}\ (n{\text{ even}}),\ P^{\mathrm {T} }QP={\begin{bmatrix}R_{1}&&&\\&\ddots &&\\&&R_{k}&\\&&&1\end{bmatrix}}\ (n{\text{ odd}}).}"></span> </p><p>where the matrices <span class="texhtml"><i>R</i><sub>1</sub>, ..., <i>R</i><sub><i>k</i></sub></span> are <span class="nowrap">2 × 2</span> rotation matrices, and with the remaining entries zero. Exceptionally, a rotation block may be diagonal, <span class="texhtml">±<i>I</i></span>. Thus, negating one column if necessary, and noting that a <span class="nowrap">2 × 2</span> reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>±</mo> <mn>1</mn> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>±</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48acff9a2be4fb4b25b3dc63554d943592143667" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:45.107ex; height:22.509ex;" alt="{\displaystyle P^{\mathrm {T} }QP={\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}"></span> </p><p>The matrices <span class="texhtml"><i>R</i><sub>1</sub>, ..., <i>R</i><sub><i>k</i></sub></span> give conjugate pairs of eigenvalues lying on the unit circle in the <a href="/wiki/Complex_number" title="Complex number">complex plane</a>; so this decomposition confirms that all <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a> have <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> 1. If <span class="texhtml mvar" style="font-style:italic;">n</span> is odd, there is at least one real eigenvalue, +1 or −1; for a <span class="nowrap">3 × 3</span> rotation, the eigenvector associated with +1 is the rotation axis. </p> <div class="mw-heading mw-heading3"><h3 id="Lie_algebra">Lie algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=11" title="Edit section: Lie algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose the entries of <span class="texhtml mvar" style="font-style:italic;">Q</span> are differentiable functions of <span class="texhtml mvar" style="font-style:italic;">t</span>, and that <span class="texhtml"><i>t</i> = 0</span> gives <span class="texhtml"><i>Q</i> = <i>I</i></span>. Differentiating the orthogonality condition <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{\mathrm {T} }Q=I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> <mo>=</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{\mathrm {T} }Q=I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8c59b38159f56ce9ae0bf267460a83ca4b8710" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.366ex; height:3.009ex;" alt="{\displaystyle Q^{\mathrm {T} }Q=I}"></span> yields <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}^{\mathrm {T} }Q+Q^{\mathrm {T} }{\dot {Q}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>Q</mi> <mo>+</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}^{\mathrm {T} }Q+Q^{\mathrm {T} }{\dot {Q}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61b909bde959dc3d84ec98f8d072d7d7d7a6ebdf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.292ex; height:3.509ex;" alt="{\displaystyle {\dot {Q}}^{\mathrm {T} }Q+Q^{\mathrm {T} }{\dot {Q}}=0}"></span> </p><p>Evaluation at <span class="texhtml"><i>t</i> = 0</span> (<span class="texhtml"><i>Q</i> = <i>I</i></span>) then implies <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Q}}^{\mathrm {T} }=-{\dot {Q}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Q}}^{\mathrm {T} }=-{\dot {Q}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacaf1e00697bce8b8f457d4d09c16d885474df1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.649ex; height:3.509ex;" alt="{\displaystyle {\dot {Q}}^{\mathrm {T} }=-{\dot {Q}}.}"></span> </p><p>In Lie group terms, this means that the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of an orthogonal matrix group consists of <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric matrices</a>. Going the other direction, the <a href="/wiki/Matrix_exponential" title="Matrix exponential">matrix exponential</a> of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal). </p><p>For example, the three-dimensional object physics calls <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> is a differential rotation, thus a vector in the Lie 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 {\mathfrak {so}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4f1d3d3bf3da64b92af1a1018ce00545808b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(3)}"></span> tangent to <span class="texhtml">SO(3)</span>. Given <span class="texhtml"><b>ω</b> = (<i>xθ</i>, <i>yθ</i>, <i>zθ</i>)</span>, with <span class="texhtml"><b>v</b> = (<i>x</i>, <i>y</i>, <i>z</i>)</span> being a unit vector, the correct skew-symmetric matrix form of <span class="texhtml mvar" style="font-style:italic;"><b>ω</b></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ={\begin{bmatrix}0&-z\theta &y\theta \\z\theta &0&-x\theta \\-y\theta &x\theta &0\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>z</mi> <mi>θ</mi> </mtd> <mtd> <mi>y</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mi>θ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>x</mi> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>y</mi> <mi>θ</mi> </mtd> <mtd> <mi>x</mi> <mi>θ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ={\begin{bmatrix}0&-z\theta &y\theta \\z\theta &0&-x\theta \\-y\theta &x\theta &0\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5330b544e12714a5dc18494b17bea5a00444a4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.19ex; height:9.509ex;" alt="{\displaystyle \Omega ={\begin{bmatrix}0&-z\theta &y\theta \\z\theta &0&-x\theta \\-y\theta &x\theta &0\end{bmatrix}}.}"></span> </p><p>The exponential of this is the orthogonal matrix for rotation around axis <span class="texhtml"><b>v</b></span> by angle <span class="texhtml mvar" style="font-style:italic;">θ</span>; setting <span class="texhtml"><i>c</i> = cos <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>θ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, <span class="texhtml"><i>s</i> = sin <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>θ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(\Omega )={\begin{bmatrix}1-2s^{2}+2x^{2}s^{2}&2xys^{2}-2zsc&2xzs^{2}+2ysc\\2xys^{2}+2zsc&1-2s^{2}+2y^{2}s^{2}&2yzs^{2}-2xsc\\2xzs^{2}-2ysc&2yzs^{2}+2xsc&1-2s^{2}+2z^{2}s^{2}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mi>x</mi> <mi>y</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mi>s</mi> <mi>c</mi> </mtd> <mtd> <mn>2</mn> <mi>x</mi> <mi>z</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mi>s</mi> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mi>y</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mi>s</mi> <mi>c</mi> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mi>y</mi> <mi>z</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>s</mi> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mi>z</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mi>s</mi> <mi>c</mi> </mtd> <mtd> <mn>2</mn> <mi>y</mi> <mi>z</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>s</mi> <mi>c</mi> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(\Omega )={\begin{bmatrix}1-2s^{2}+2x^{2}s^{2}&2xys^{2}-2zsc&2xzs^{2}+2ysc\\2xys^{2}+2zsc&1-2s^{2}+2y^{2}s^{2}&2yzs^{2}-2xsc\\2xzs^{2}-2ysc&2yzs^{2}+2xsc&1-2s^{2}+2z^{2}s^{2}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd059b06ddd6a4ff7d038a1cf38c3812572521f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.973ex; margin-bottom: -0.198ex; width:66.398ex; height:9.509ex;" alt="{\displaystyle \exp(\Omega )={\begin{bmatrix}1-2s^{2}+2x^{2}s^{2}&2xys^{2}-2zsc&2xzs^{2}+2ysc\\2xys^{2}+2zsc&1-2s^{2}+2y^{2}s^{2}&2yzs^{2}-2xsc\\2xzs^{2}-2ysc&2yzs^{2}+2xsc&1-2s^{2}+2z^{2}s^{2}\end{bmatrix}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Numerical_linear_algebra">Numerical linear algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=12" title="Edit section: Numerical linear algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Benefits">Benefits</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=13" title="Edit section: Benefits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices. Having determinant ±1 and all eigenvalues of magnitude 1 is of great benefit for <a href="/wiki/Numeric_stability" class="mw-redirect" title="Numeric stability">numeric stability</a>. One implication is that the <a href="/wiki/Condition_number" title="Condition number">condition number</a> is 1 (which is the minimum), so errors are not magnified when multiplying with an orthogonal matrix. Many algorithms use orthogonal matrices like Householder reflections and <a href="/wiki/Givens_rotation" title="Givens rotation">Givens rotations</a> for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices. </p><p>Permutations are essential to the success of many algorithms, including the workhorse <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> with <a href="/wiki/Pivot_element#Partial_and_complete_pivoting" title="Pivot element">partial pivoting</a> (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of <span class="texhtml mvar" style="font-style:italic;">n</span> indices. </p><p>Likewise, algorithms using Householder and Givens matrices typically use specialized methods of multiplication and storage. For example, a Givens rotation affects only two rows of a matrix it multiplies, changing a full <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">multiplication</a> of order <span class="texhtml"><i>n</i><sup>3</sup></span> to a much more efficient order <span class="texhtml mvar" style="font-style:italic;">n</span>. When uses of these reflections and rotations introduce zeros in a matrix, the space vacated is enough to store sufficient data to reproduce the transform, and to do so robustly. (Following <a href="#CITEREFStewart1976">Stewart (1976)</a>, we do <i>not</i> store a rotation angle, which is both expensive and badly behaved.) </p> <div class="mw-heading mw-heading3"><h3 id="Decompositions">Decompositions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=14" title="Edit section: Decompositions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A number of important <a href="/wiki/Matrix_decomposition" title="Matrix decomposition">matrix decompositions</a> (<a href="#CITEREFGolubVan_Loan1996">Golub & Van Loan 1996</a>) involve orthogonal matrices, including especially: </p> <dl><dt><a href="/wiki/QR_decomposition" title="QR decomposition"><span class="texhtml mvar" style="font-style:italic;">QR</span> decomposition</a></dt> <dd><span class="texhtml"><i>M</i> = <i>QR</i></span>, <span class="texhtml mvar" style="font-style:italic;">Q</span> orthogonal, <span class="texhtml mvar" style="font-style:italic;">R</span> upper triangular</dd> <dt><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></dt> <dd><span class="texhtml"><i>M</i> = <i>U</i>Σ<i>V</i><sup>T</sup></span>, <span class="texhtml mvar" style="font-style:italic;">U</span> and <span class="texhtml mvar" style="font-style:italic;">V</span> orthogonal, <span class="texhtml">Σ</span> diagonal matrix</dd> <dt><a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">Eigendecomposition of a symmetric matrix</a> (decomposition according to the <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a>)</dt> <dd><span class="texhtml"><i>S</i> = <i>Q</i>Λ<i>Q</i><sup>T</sup></span>, <span class="texhtml mvar" style="font-style:italic;">S</span> symmetric, <span class="texhtml mvar" style="font-style:italic;">Q</span> orthogonal, <span class="texhtml">Λ</span> diagonal</dd> <dt><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></dt> <dd><span class="texhtml"><i>M</i> = <i>QS</i></span>, <span class="texhtml mvar" style="font-style:italic;">Q</span> orthogonal, <span class="texhtml mvar" style="font-style:italic;">S</span> symmetric positive-semidefinite</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Examples_2">Examples</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=15" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider an <a href="/wiki/Overdetermined_system_of_linear_equations" class="mw-redirect" title="Overdetermined system of linear equations">overdetermined system of linear equations</a>, as might occur with repeated measurements of a physical phenomenon to compensate for experimental errors. Write <span class="texhtml"><i>A</i><b>x</b> = <b>b</b></span>, where <span class="texhtml mvar" style="font-style:italic;">A</span> is <span class="texhtml"><i>m</i> × <i>n</i></span>, <span class="texhtml"><i>m</i> > <i>n</i></span>. A <span class="texhtml mvar" style="font-style:italic;">QR</span> decomposition reduces <span class="texhtml mvar" style="font-style:italic;">A</span> to upper triangular <span class="texhtml mvar" style="font-style:italic;">R</span>. For example, if <span class="texhtml mvar" style="font-style:italic;">A</span> is <span class="nowrap">5 × 3</span> then <span class="texhtml mvar" style="font-style:italic;">R</span> has the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\begin{bmatrix}\cdot &\cdot &\cdot \\0&\cdot &\cdot \\0&0&\cdot \\0&0&0\\0&0&0\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>⋅</mo> </mtd> <mtd> <mo>⋅</mo> </mtd> <mtd> <mo>⋅</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </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> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\begin{bmatrix}\cdot &\cdot &\cdot \\0&\cdot &\cdot \\0&0&\cdot \\0&0&0\\0&0&0\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f16b007da3153faeb457c0894f402b70ec8caaca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:17.494ex; height:15.843ex;" alt="{\displaystyle R={\begin{bmatrix}\cdot &\cdot &\cdot \\0&\cdot &\cdot \\0&0&\cdot \\0&0&0\\0&0&0\end{bmatrix}}.}"></span> </p><p>The <a href="/wiki/Linear_least_squares_(mathematics)" class="mw-redirect" title="Linear least squares (mathematics)">linear least squares</a> problem is to find the <span class="texhtml"><b>x</b></span> that minimizes <span class="texhtml">‖<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>A</i><b>x</b> − <b>b</b></span>‖</span>, which is equivalent to projecting <span class="texhtml"><b>b</b></span> to the subspace spanned by the columns of <span class="texhtml mvar" style="font-style:italic;">A</span>. Assuming the columns of <span class="texhtml mvar" style="font-style:italic;">A</span> (and hence <span class="texhtml mvar" style="font-style:italic;">R</span>) are independent, the projection solution is found from <span class="texhtml"><i>A</i><sup>T</sup><i>A</i><b>x</b> = <i>A</i><sup>T</sup><b>b</b></span>. Now <span class="texhtml"><i>A</i><sup>T</sup><i>A</i></span> is square (<span class="texhtml"><i>n</i> × <i>n</i></span>) and invertible, and also equal to <span class="texhtml"><i>R</i><sup>T</sup><i>R</i></span>. But the lower rows of zeros in <span class="texhtml mvar" style="font-style:italic;">R</span> are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> (<a href="/wiki/Cholesky_decomposition" title="Cholesky decomposition">Cholesky decomposition</a>). Here orthogonality is important not only for reducing <span class="texhtml"><i>A</i><sup>T</sup><i>A</i> = (<i>R</i><sup>T</sup><i>Q</i><sup>T</sup>)<i>QR</i></span> to <span class="texhtml"><i>R</i><sup>T</sup><i>R</i></span>, but also for allowing solution without magnifying numerical problems. </p><p>In the case of a linear system which is underdetermined, or an otherwise non-<a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible matrix</a>, singular value decomposition (SVD) is equally useful. With <span class="texhtml mvar" style="font-style:italic;">A</span> factored as <span class="texhtml"><i>U</i>Σ<i>V</i><sup>T</sup></span>, a satisfactory solution uses the Moore-Penrose <a href="/wiki/Pseudoinverse" class="mw-redirect" title="Pseudoinverse">pseudoinverse</a>, <span class="texhtml"><i>V</i>Σ<sup>+</sup><i>U</i><sup>T</sup></span>, where <span class="texhtml">Σ<sup>+</sup></span> merely replaces each non-zero diagonal entry with its reciprocal. Set <span class="texhtml"><b>x</b></span> to <span class="texhtml"><i>V</i>Σ<sup>+</sup><i>U</i><sup>T</sup><b>b</b></span>. </p><p>The case of a square invertible matrix also holds interest. Suppose, for example, that <span class="texhtml mvar" style="font-style:italic;">A</span> is a <span class="nowrap">3 × 3</span> rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so <span class="texhtml mvar" style="font-style:italic;">A</span> has gradually lost its true orthogonality. A <a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a> could <a href="/wiki/Orthogonalization" title="Orthogonalization">orthogonalize</a> the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The <a href="/wiki/Polar_decomposition" title="Polar decomposition">polar decomposition</a> factors a matrix into a pair, one of which is the unique <i>closest</i> orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any <a href="/wiki/Matrix_norm" title="Matrix norm">matrix norm</a> invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "<a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>" approach due to <a href="#CITEREFHigham1986">Higham (1986)</a> (<a href="#CITEREFHigham1990">1990</a>), repeatedly averaging the matrix with its inverse transpose. <a href="#CITEREFDubrulle1999">Dubrulle (1999)</a> has published an accelerated method with a convenient convergence test. </p><p>For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.8125&0.0625\\3.4375&2.6875\end{bmatrix}}\rightarrow \cdots \rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1.8125</mn> </mtd> <mtd> <mn>0.0625</mn> </mtd> </mtr> <mtr> <mtd> <mn>3.4375</mn> </mtd> <mtd> <mn>2.6875</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mo>⋯</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0.8</mn> </mtd> <mtd> <mo>−</mo> <mn>0.6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.6</mn> </mtd> <mtd> <mn>0.8</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.8125&0.0625\\3.4375&2.6875\end{bmatrix}}\rightarrow \cdots \rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3190976cdd7626ac06ec3aec4ca81de3a5f39713" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.147ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.8125&0.0625\\3.4375&2.6875\end{bmatrix}}\rightarrow \cdots \rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}"></span> and which acceleration trims to two steps (with <span class="texhtml mvar" style="font-style:italic;">γ</span> = 0.353553, 0.565685). </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.41421&-1.06066\\1.06066&1.41421\end{bmatrix}}\rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1.41421</mn> </mtd> <mtd> <mo>−</mo> <mn>1.06066</mn> </mtd> </mtr> <mtr> <mtd> <mn>1.06066</mn> </mtd> <mtd> <mn>1.41421</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0.8</mn> </mtd> <mtd> <mo>−</mo> <mn>0.6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.6</mn> </mtd> <mtd> <mn>0.8</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.41421&-1.06066\\1.06066&1.41421\end{bmatrix}}\rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddd7b89537d8aa4ecd2f921712dee00511608e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.943ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}1.41421&-1.06066\\1.06066&1.41421\end{bmatrix}}\rightarrow {\begin{bmatrix}0.8&-0.6\\0.6&0.8\end{bmatrix}}}"></span> </p><p>Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}0.393919&-0.919145\\0.919145&0.393919\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0.393919</mn> </mtd> <mtd> <mo>−</mo> <mn>0.919145</mn> </mtd> </mtr> <mtr> <mtd> <mn>0.919145</mn> </mtd> <mtd> <mn>0.393919</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}0.393919&-0.919145\\0.919145&0.393919\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e127c2f15c68e59e205641bba37dd96999b9d62" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.373ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}3&1\\7&5\end{bmatrix}}\rightarrow {\begin{bmatrix}0.393919&-0.919145\\0.919145&0.393919\end{bmatrix}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Randomization">Randomization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=16" title="Edit section: Randomization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some numerical applications, such as <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo methods</a> and exploration of high-dimensional data spaces, require generation of <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniformly distributed</a> random orthogonal matrices. In this context, "uniform" is defined in terms of <a href="/wiki/Haar_measure" title="Haar measure">Haar measure</a>, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. Orthogonalizing matrices with <a href="/wiki/Statistical_independence" class="mw-redirect" title="Statistical independence">independent</a> uniformly distributed random entries does not result in uniformly distributed orthogonal matrices<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2009)">citation needed</span></a></i>]</sup>, but the <a href="/wiki/QR_decomposition" title="QR decomposition"><span class="texhtml mvar" style="font-style:italic;">QR</span> decomposition</a> of independent <a href="/wiki/Normal_distribution" title="Normal distribution">normally distributed</a> random entries does, as long as the diagonal of <span class="texhtml mvar" style="font-style:italic;">R</span> contains only positive entries (<a href="#CITEREFMezzadri2006">Mezzadri 2006</a>). <a href="#CITEREFStewart1980">Stewart (1980)</a> replaced this with a more efficient idea that <a href="#CITEREFDiaconisShahshahani1987">Diaconis & Shahshahani (1987)</a> later generalized as the "subgroup algorithm" (in which form it works just as well for permutations and rotations). To generate an <span class="texhtml">(<i>n</i> + 1) × (<i>n</i> + 1)</span> orthogonal matrix, take an <span class="texhtml"><i>n</i> × <i>n</i></span> one and a uniformly distributed unit vector of dimension <span class="nowrap"><i>n</i> + 1</span>. Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner). </p> <div class="mw-heading mw-heading3"><h3 id="Nearest_orthogonal_matrix">Nearest orthogonal matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=17" title="Edit section: Nearest orthogonal matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The problem of finding the orthogonal matrix <span class="texhtml mvar" style="font-style:italic;">Q</span> nearest a given matrix <span class="texhtml mvar" style="font-style:italic;">M</span> is related to the <a href="/wiki/Orthogonal_Procrustes_problem" title="Orthogonal Procrustes problem">Orthogonal Procrustes problem</a>. There are several different ways to get the unique solution, the simplest of which is taking the <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> of <span class="texhtml mvar" style="font-style:italic;">M</span> and replacing the singular values with ones. Another method expresses the <span class="texhtml mvar" style="font-style:italic;">R</span> explicitly but requires the use of a <a href="/wiki/Matrix_square_root" class="mw-redirect" title="Matrix square root">matrix square root</a>:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=M\left(M^{\mathrm {T} }M\right)^{-{\frac {1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>M</mi> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mi>M</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=M\left(M^{\mathrm {T} }M\right)^{-{\frac {1}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be490fed1664c021f534e44e07c706ef7feadeb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.883ex; height:4.509ex;" alt="{\displaystyle Q=M\left(M^{\mathrm {T} }M\right)^{-{\frac {1}{2}}}}"></span> </p><p>This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n+1}=2M\left(Q_{n}^{-1}M+M^{\mathrm {T} }Q_{n}\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>M</mi> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mi>M</mi> <mo>+</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n+1}=2M\left(Q_{n}^{-1}M+M^{\mathrm {T} }Q_{n}\right)^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4c120980140cb32a74691ea3fa5b1672aec376" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.751ex; height:3.843ex;" alt="{\displaystyle Q_{n+1}=2M\left(Q_{n}^{-1}M+M^{\mathrm {T} }Q_{n}\right)^{-1}}"></span> where <span class="texhtml"><i>Q</i><sub>0</sub> = <i>M</i></span>. </p><p>These iterations are stable provided the <a href="/wiki/Condition_number" title="Condition number">condition number</a> of <span class="texhtml mvar" style="font-style:italic;">M</span> is less than three.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Using a first-order approximation of the inverse and the same initialization results in the modified iteration: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{n}=Q_{n}^{\mathrm {T} }Q_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msubsup> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{n}=Q_{n}^{\mathrm {T} }Q_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca0f275829df2d5c392649508c695ec9b3bf659" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.497ex; height:2.843ex;" alt="{\displaystyle N_{n}=Q_{n}^{\mathrm {T} }Q_{n}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}={\frac {1}{2}}Q_{n}N_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}={\frac {1}{2}}Q_{n}N_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d49d32132089c440b7031d0b4fffd27cb9ec1546" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.949ex; height:5.176ex;" alt="{\displaystyle P_{n}={\frac {1}{2}}Q_{n}N_{n}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n+1}=2Q_{n}+P_{n}N_{n}-3P_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mn>3</mn> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n+1}=2Q_{n}+P_{n}N_{n}-3P_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce061cbec85e11f4da27f6318b53e8af6deaf398" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.824ex; height:2.509ex;" alt="{\displaystyle Q_{n+1}=2Q_{n}+P_{n}N_{n}-3P_{n}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Spin_and_pin">Spin and pin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=18" title="Edit section: Spin and pin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not <a href="/wiki/Connected_space" title="Connected space">connected</a> to each other, even the +1 component, <span class="texhtml">SO(<i>n</i>)</span>, is not <a href="/wiki/Simply_connected_space" title="Simply connected space">simply connected</a> (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a <a href="/wiki/Covering_map" class="mw-redirect" title="Covering map">covering group</a> of SO(<i>n</i>), the <a href="/wiki/Spinor_group" class="mw-redirect" title="Spinor group">spin group</a>, <span class="texhtml">Spin(<i>n</i>)</span>. Likewise, <span class="texhtml">O(<i>n</i>)</span> has covering groups, the <a href="/wiki/Pin_group" title="Pin group">pin groups</a>, Pin(<i>n</i>). For <span class="texhtml"><i>n</i> > 2</span>, <span class="texhtml">Spin(<i>n</i>)</span> is simply connected and thus the universal covering group for <span class="texhtml">SO(<i>n</i>)</span>. By far the most famous example of a spin group is <span class="texhtml">Spin(3)</span>, which is nothing but <span class="texhtml">SU(2)</span>, or the group of unit <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>. </p><p>The Pin and Spin groups are found within <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a>, which themselves can be built from orthogonal matrices. </p> <div class="mw-heading mw-heading2"><h2 id="Rectangular_matrices">Rectangular matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=19" title="Edit section: Rectangular matrices"><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/Semi-orthogonal_matrix" title="Semi-orthogonal matrix">Semi-orthogonal matrix</a></div> <p>If <span class="texhtml mvar" style="font-style:italic;">Q</span> is not a square matrix, then the conditions <span class="texhtml"><i>Q</i><sup>T</sup><i>Q</i> = <i>I</i></span> and <span class="texhtml"><i>QQ</i><sup>T</sup> = <i>I</i></span> are not equivalent. The condition <span class="texhtml"><i>Q</i><sup>T</sup><i>Q</i> = <i>I</i></span> says that the columns of <i>Q</i> are orthonormal. This can only happen if <span class="texhtml mvar" style="font-style:italic;">Q</span> is an <span class="texhtml"><i>m</i> × <i>n</i></span> matrix with <span class="texhtml"><i>n</i> ≤ <i>m</i></span> (due to linear dependence). Similarly, <span class="texhtml"><i>QQ</i><sup>T</sup> = <i>I</i></span> says that the rows of <span class="texhtml mvar" style="font-style:italic;">Q</span> are orthonormal, which requires <span class="texhtml"><i>n</i> ≥ <i>m</i></span>. </p><p>There is no standard terminology for these matrices. They are variously called "semi-orthogonal matrices", "orthonormal matrices", "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns". </p><p>For the case <span class="texhtml"><i>n</i> ≤ <i>m</i></span>, matrices with orthonormal columns may be referred to as <a href="/wiki/K-frame" title="K-frame">orthogonal k-frames</a> and they are elements of the <a href="/wiki/Stiefel_manifold" title="Stiefel manifold">Stiefel manifold</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=Orthogonal_matrix&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Biorthogonal_system" title="Biorthogonal system">Biorthogonal system</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=21" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://tutorial.math.lamar.edu/Classes/LinAlg/OrthogonalMatrix.aspx">"Paul's online math notes"</a><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (January 2013)">full citation needed</span></a></i>]</sup>, Paul Dawkins, <a href="/wiki/Lamar_University" title="Lamar University">Lamar University</a>, 2008. Theorem 3(c)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://people.csail.mit.edu/bkph/articles/Nearest_Orthonormal_Matrix.pdf">"Finding the Nearest Orthonormal Matrix"</a>, <a href="/wiki/Berthold_K.P._Horn" title="Berthold K.P. Horn">Berthold K.P. Horn</a>, <a href="/wiki/MIT" class="mw-redirect" title="MIT">MIT</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.maths.manchester.ac.uk/~nareports/narep91.pdf">"Newton's Method for the Matrix Square Root"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110929131330/http://www.maths.manchester.ac.uk/~nareports/narep91.pdf">Archived</a> 2011-09-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Nicholas J. Higham, Mathematics of Computation, Volume 46, Number 174, 1986.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFDiaconisShahshahani1987" class="citation cs2"><a href="/wiki/Persi_Diaconis" title="Persi Diaconis">Diaconis, Persi</a>; Shahshahani, Mehrdad (1987), "The subgroup algorithm for generating uniform random variables", <i>Probability in the Engineering and Informational Sciences</i>, <b>1</b>: 15–32, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0269964800000255">10.1017/S0269964800000255</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0269-9648">0269-9648</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122752374">122752374</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Probability+in+the+Engineering+and+Informational+Sciences&rft.atitle=The+subgroup+algorithm+for+generating+uniform+random+variables&rft.volume=1&rft.pages=15-32&rft.date=1987&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122752374%23id-name%3DS2CID&rft.issn=0269-9648&rft_id=info%3Adoi%2F10.1017%2FS0269964800000255&rft.aulast=Diaconis&rft.aufirst=Persi&rft.au=Shahshahani%2C+Mehrdad&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDubrulle1999" class="citation cs2">Dubrulle, Augustin A. (1999), <a rel="nofollow" class="external text" href="http://etna.mcs.kent.edu/">"An Optimum Iteration for the Matrix Polar Decomposition"</a>, <i>Electronic Transactions on Numerical Analysis</i>, <b>8</b>: 21–25</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Transactions+on+Numerical+Analysis&rft.atitle=An+Optimum+Iteration+for+the+Matrix+Polar+Decomposition&rft.volume=8&rft.pages=21-25&rft.date=1999&rft.aulast=Dubrulle&rft.aufirst=Augustin+A.&rft_id=http%3A%2F%2Fetna.mcs.kent.edu%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubVan_Loan1996" class="citation cs2"><a href="/wiki/Gene_H._Golub" title="Gene H. Golub">Golub, Gene H.</a>; <a href="/wiki/Charles_F._Van_Loan" title="Charles F. 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W.</a> (1976), "The Economical Storage of Plane Rotations", <i>Numerische Mathematik</i>, <b>25</b> (2): 137–138, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01462266">10.1007/BF01462266</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0029-599X">0029-599X</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120372682">120372682</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Numerische+Mathematik&rft.atitle=The+Economical+Storage+of+Plane+Rotations&rft.volume=25&rft.issue=2&rft.pages=137-138&rft.date=1976&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120372682%23id-name%3DS2CID&rft.issn=0029-599X&rft_id=info%3Adoi%2F10.1007%2FBF01462266&rft.aulast=Stewart&rft.aufirst=G.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewart1980" class="citation cs2"><a href="/w/index.php?title=G._W._Stewart&action=edit&redlink=1" class="new" title="G. 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W.</a> (1980), "The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators", <i>SIAM Journal on Numerical Analysis</i>, <b>17</b> (3): 403–409, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1980SJNA...17..403S">1980SJNA...17..403S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0717034">10.1137/0717034</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0036-1429">0036-1429</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Numerical+Analysis&rft.atitle=The+Efficient+Generation+of+Random+Orthogonal+Matrices+with+an+Application+to+Condition+Estimators&rft.volume=17&rft.issue=3&rft.pages=403-409&rft.date=1980&rft.issn=0036-1429&rft_id=info%3Adoi%2F10.1137%2F0717034&rft_id=info%3Abibcode%2F1980SJNA...17..403S&rft.aulast=Stewart&rft.aufirst=G.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMezzadri2006" class="citation cs2">Mezzadri, Francesco (2006), "How to generate random matrices from the classical compact groups", <i>Notices of the American Mathematical Society</i>, <b>54</b>, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math-ph/0609050">math-ph/0609050</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006math.ph...9050M">2006math.ph...9050M</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=How+to+generate+random+matrices+from+the+classical+compact+groups&rft.volume=54&rft.date=2006&rft_id=info%3Aarxiv%2Fmath-ph%2F0609050&rft_id=info%3Abibcode%2F2006math.ph...9050M&rft.aulast=Mezzadri&rft.aufirst=Francesco&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+matrix" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orthogonal_matrix&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist"><a href="https://en.wikiversity.org/wiki/Linear_algebra/Orthogonal_matrix" class="extiw" title="v:Linear algebra/Orthogonal matrix">Wikiversity introduces the <b>orthogonal matrix</b>.</a></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Orthogonal_matrix">"Orthogonal matrix"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Orthogonal+matrix&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DOrthogonal_matrix&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOrthogonal+matrix" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://people.revoledu.com/kardi/tutorial/LinearAlgebra/MatrixOrthogonal.html">Tutorial and Interactive Program on Orthogonal Matrix</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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.navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Matrix_classes" title="Template:Matrix classes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Matrix_classes" title="Template talk:Matrix classes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Matrix_classes" title="Special:EditPage/Template:Matrix classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Matrix_classes" style="font-size:114%;margin:0 4em"><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" 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 href="/wiki/Triangular_matrix" title="Triangular matrix">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 class="mw-selflink selflink">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" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li> <li><a href="/wiki/List_of_matrices" class="mw-redirect" title="List of matrices">List of matrices</a></li> <li><a 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Orthogonal matrix - Wikipedia