Trace (linear algebra)

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href="#Properties"> <div class="vector-toc-text"> 3 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Basic_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basic_properties"> <div class="vector-toc-text"> 3.1 Basic properties </div> </a> <ul id="toc-Basic_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trace_of_a_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trace_of_a_product"> <div class="vector-toc-text"> 3.2 Trace of a product </div> </a> <ul id="toc-Trace_of_a_product-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cyclic_property" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cyclic_property"> <div class="vector-toc-text"> 3.3 Cyclic property </div> </a> <ul id="toc-Cyclic_property-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trace_of_a_Kronecker_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trace_of_a_Kronecker_product"> <div class="vector-toc-text"> 3.4 Trace of a Kronecker product </div> </a> <ul id="toc-Trace_of_a_Kronecker_product-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characterization_of_the_trace" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characterization_of_the_trace"> <div class="vector-toc-text"> 3.5 Characterization of the trace </div> </a> <ul id="toc-Characterization_of_the_trace-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trace_as_the_sum_of_eigenvalues" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trace_as_the_sum_of_eigenvalues"> <div class="vector-toc-text"> 3.6 Trace as the sum of eigenvalues </div> </a> <ul id="toc-Trace_as_the_sum_of_eigenvalues-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trace_of_commutator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trace_of_commutator"> <div class="vector-toc-text"> 3.7 Trace of commutator </div> </a> <ul id="toc-Trace_of_commutator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Traces_of_special_kinds_of_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Traces_of_special_kinds_of_matrices"> <div class="vector-toc-text"> 3.8 Traces of special kinds of matrices </div> </a> <ul id="toc-Traces_of_special_kinds_of_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_the_characteristic_polynomial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_the_characteristic_polynomial"> <div class="vector-toc-text"> 3.9 Relationship to the characteristic polynomial </div> </a> <ul id="toc-Relationship_to_the_characteristic_polynomial-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relationship_to_eigenvalues" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_to_eigenvalues"> <div class="vector-toc-text"> 4 Relationship to eigenvalues </div> </a> <button aria-controls="toc-Relationship_to_eigenvalues-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Relationship to eigenvalues subsection </button> <ul id="toc-Relationship_to_eigenvalues-sublist" class="vector-toc-list"> <li id="toc-Derivative_relationships" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivative_relationships"> <div class="vector-toc-text"> 4.1 Derivative relationships </div> </a> <ul id="toc-Derivative_relationships-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Trace_of_a_linear_operator" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Trace_of_a_linear_operator"> <div class="vector-toc-text"> 5 Trace of a linear operator </div> </a> <ul id="toc-Trace_of_a_linear_operator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical_algorithms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Numerical_algorithms"> <div class="vector-toc-text"> 6 Numerical algorithms </div> </a> <button aria-controls="toc-Numerical_algorithms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Numerical algorithms subsection </button> <ul id="toc-Numerical_algorithms-sublist" class="vector-toc-list"> <li id="toc-Stochastic_estimator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stochastic_estimator"> <div class="vector-toc-text"> 6.1 Stochastic estimator </div> </a> <ul id="toc-Stochastic_estimator-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 7 Applications </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lie_algebra"> <div class="vector-toc-text"> 8 Lie algebra </div> </a> <button aria-controls="toc-Lie_algebra-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Lie algebra subsection </button> <ul id="toc-Lie_algebra-sublist" class="vector-toc-list"> <li id="toc-Bilinear_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bilinear_forms"> <div class="vector-toc-text"> 8.1 Bilinear forms </div> </a> <ul id="toc-Bilinear_forms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 9 Generalizations </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Traces_in_the_language_of_tensor_products" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Traces_in_the_language_of_tensor_products"> <div class="vector-toc-text"> 10 Traces in the language of tensor products </div> </a> <ul id="toc-Traces_in_the_language_of_tensor_products-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"> 11 See also </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"> 12 Notes </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"> 13 References </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"> 14 External links </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" > Toggle the table of contents </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">Trace (linear algebra)</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 40 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-40" aria-hidden="true" > 40 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A3%D8%AB%D8%B1_(%D8%AC%D8%A8%D8%B1_%D8%AE%D8%B7%D9%8A)" title="أثر (جبر خطي) – Arabic" lang="ar" hreflang="ar" data-title="أثر (جبر خطي)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Jiah_(S%C3%B2a%E2%81%BF-s%C3%A8ng_t%C4%81i-s%C3%B2%CD%98)" title="Jiah (Sòaⁿ-sèng tāi-sò͘) – Minnan" lang="nan" hreflang="nan" data-title="Jiah (Sòaⁿ-sèng tāi-sò͘)" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Tra%C3%A7a_(%C3%A0lgebra_lineal)" title="Traça (àlgebra lineal) – Catalan" lang="ca" hreflang="ca" data-title="Traça (àlgebra lineal)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_%D0%B9%C4%95%D1%80%C4%95" title="Матрица йĕрĕ – Chuvash" lang="cv" hreflang="cv" data-title="Матрица йĕрĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Stopa_matice" title="Stopa matice – Czech" lang="cs" hreflang="cs" data-title="Stopa matice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Spor_(algebra)" title="Spor (algebra) – Danish" lang="da" hreflang="da" data-title="Spor (algebra)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Spur_(Mathematik)" title="Spur (Mathematik) – German" lang="de" hreflang="de" data-title="Spur (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/J%C3%A4lg_(lineaaralgebra)" title="Jälg (lineaaralgebra) – Estonian" lang="et" hreflang="et" data-title="Jälg (lineaaralgebra)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%8A%CF%87%CE%BD%CE%BF%CF%82_%CF%80%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1" title="Ίχνος πίνακα – Greek" lang="el" hreflang="el" data-title="Ίχνος πίνακα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Traza_(%C3%A1lgebra_lineal)" title="Traza (álgebra lineal) – Spanish" lang="es" hreflang="es" data-title="Traza (álgebra lineal)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Spuro_(lineara_algebro)" title="Spuro (lineara algebro) – Esperanto" lang="eo" hreflang="eo" data-title="Spuro (lineara algebro)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aztarna_(aljebra)" title="Aztarna (aljebra) – Basque" lang="eu" hreflang="eu" data-title="Aztarna (aljebra)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AB%D8%B1_(%D8%AC%D8%A8%D8%B1_%D8%AE%D8%B7%DB%8C)" title="اثر (جبر خطی) – Persian" lang="fa" hreflang="fa" data-title="اثر (جبر خطی)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Trace_(alg%C3%A8bre)" title="Trace (algèbre) – French" lang="fr" hreflang="fr" data-title="Trace (algèbre)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Traza_(%C3%A1lxebra_linear)" title="Traza (álxebra linear) – Galician" lang="gl" hreflang="gl" data-title="Traza (álxebra linear)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EA%B0%81%ED%95%A9" title="대각합 – Korean" lang="ko" hreflang="ko" data-title="대각합" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teras_(aljabar_linear)" title="Teras (aljabar linear) – Indonesian" lang="id" hreflang="id" data-title="Teras (aljabar linear)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Traccia_(matrice)" title="Traccia (matrice) – Italian" lang="it" hreflang="it" data-title="Traccia (matrice)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A7%D7%91%D7%94_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="עקבה (אלגברה) – Hebrew" lang="he" hreflang="he" data-title="עקבה (אלגברה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Nyom_(line%C3%A1ris_algebra)" title="Nyom (lineáris algebra) – Hungarian" lang="hu" hreflang="hu" data-title="Nyom (lineáris algebra)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Spoor_(lineaire_algebra)" title="Spoor (lineaire algebra) – Dutch" lang="nl" hreflang="nl" data-title="Spoor (lineaire algebra)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%B7%A1_(%E7%B7%9A%E5%9E%8B%E4%BB%A3%E6%95%B0%E5%AD%A6)" title="跡 (線型代数学) – Japanese" lang="ja" hreflang="ja" data-title="跡 (線型代数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Spor_i_matematikk" title="Spor i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Spor i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%9Alad_(algebra_liniowa)" title="Ślad (algebra liniowa) – Polish" lang="pl" hreflang="pl" data-title="Ślad (algebra liniowa)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tra%C3%A7o_(%C3%A1lgebra_linear)" title="Traço (álgebra linear) – Portuguese" lang="pt" hreflang="pt" data-title="Traço (álgebra linear)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Urm%C4%83_(algebr%C4%83)" title="Urmă (algebră) – Romanian" lang="ro" hreflang="ro" data-title="Urmă (algebră)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D0%B5%D0%B4_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8B" title="След матрицы – Russian" lang="ru" hreflang="ru" data-title="След матрицы" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Gjurma_(algjeb%C3%ABr_lineare)" title="Gjurma (algjebër lineare) – Albanian" lang="sq" hreflang="sq" data-title="Gjurma (algjebër lineare)" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Stopa_matice" title="Stopa matice – Slovak" lang="sk" hreflang="sk" data-title="Stopa matice" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Sled_matrike" title="Sled matrike – Slovenian" lang="sl" hreflang="sl" data-title="Sled matrike" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D1%80%D0%B0%D0%B3_(%D0%BB%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Траг (линеарна алгебра) – Serbian" lang="sr" hreflang="sr" data-title="Траг (линеарна алгебра)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/J%C3%A4lki" title="Jälki – Finnish" lang="fi" hreflang="fi" data-title="Jälki" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sp%C3%A5r_(matematik)" title="Spår (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Spår (matematik)" data-language-autonym="Svenska" data-language-local-name="Swedish" 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mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%98%D0%B7%D0%B8_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%82%D1%81%D0%B0" title="Изи матритса – Tajik" lang="tg" hreflang="tg" data-title="Изи матритса" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0lkk%C3%B6%C5%9Fegen_toplam%C4%B1" title="İlkköşegen toplamı – Turkish" lang="tr" hreflang="tr" data-title="İlkköşegen toplamı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BB%D1%96%D0%B4_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%96" title="Слід матриці – Ukrainian" lang="uk" hreflang="uk" data-title="Слід матриці" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Sum of elements on the main diagonal</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 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class="mbox-text-span">This article needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a>. Please help <a href="/wiki/Special:EditPage/Trace_(linear_algebra)" title="Special:EditPage/Trace (linear algebra)">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed. Find sources: <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Trace%22+linear+algebra">"Trace" linear algebra</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Trace%22+linear+algebra+-wikipedia&tbs=ar:1">news</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Trace%22+linear+algebra&tbs=bkt:s&tbm=bks">newspapers</a> · <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Trace%22+linear+algebra+-wikipedia">books</a> · <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Trace%22+linear+algebra">scholar</a> · <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Trace%22+linear+algebra&acc=on&wc=on">JSTOR</a> (November 2023) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> In <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, the trace of a <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> A, denoted tr(A),<a href="#cite_note-:1-1">[1]</a> is the sum of the elements on its <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{11}+a_{22}+\dots +a_{nn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{11}+a_{22}+\dots +a_{nn}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10262aa5642e6b61e0ab87bc323be450db82424" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.891ex; height:2.343ex;" alt="{\displaystyle a_{11}+a_{22}+\dots +a_{nn}}">. It is only defined for a square matrix (n × n). In mathematical physics, if tr(A) = 0, the matrix is said to be traceless. This misnomer is widely used, as in the definition of <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>. The trace of a matrix is the sum of its <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> (counted with multiplicities). Also, tr(AB) = tr(BA) for any matrices A and B of the same size. Thus, <a href="/wiki/Matrix_similarity" title="Matrix similarity">similar matrices</a> have the same trace. As a consequence, one can define the trace of a <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> mapping a finite-dimensional <a href="/wiki/Vector_space" title="Vector space">vector space</a> into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the <a href="/wiki/Determinant" title="Determinant">determinant</a> (see <a href="/wiki/Jacobi%27s_formula" title="Jacobi's formula">Jacobi's formula</a>). <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> The trace of an n × n <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> A is defined as<a href="#cite_note-:1-1">[1]</a><a href="#cite_note-:2-2">[2]</a><a href="#cite_note-LipschutzLipson-3">[3]</a>: 34  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}a_{ii}=a_{11}+a_{22}+\dots +a_{nn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}a_{ii}=a_{11}+a_{22}+\dots +a_{nn}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627945b163d7d9a1c68d59c905de5088f2e71f31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.072ex; height:6.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}a_{ii}=a_{11}+a_{22}+\dots +a_{nn}}"> where aii denotes the entry on the i th row and i th column of A. The entries of A can be <a href="/wiki/Real_number" title="Real number">real numbers</a>, <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>, or more generally elements of a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> F. The trace is not defined for non-square matrices. <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=2" title="Edit section: Example">edit</a>]</div> Let A be a matrix, with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}={\begin{pmatrix}1&0&3\\11&5&2\\6&12&-5\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>11</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}={\begin{pmatrix}1&0&3\\11&5&2\\6&12&-5\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6988ef404cd2ffb46934efd60fdd9466612c904" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:44.082ex; height:9.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}={\begin{pmatrix}1&0&3\\11&5&2\\6&12&-5\end{pmatrix}}}"> Then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{3}a_{ii}=a_{11}+a_{22}+a_{33}=1+5+(-5)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>5</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{3}a_{ii}=a_{11}+a_{22}+a_{33}=1+5+(-5)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/076c877850a7b9ca25f1fa7f567aaaab342b8074" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.325ex; height:7.176ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{3}a_{ii}=a_{11}+a_{22}+a_{33}=1+5+(-5)=1}"> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=3" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Basic_properties">Basic properties</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=4" title="Edit section: Basic properties">edit</a>]</div> The trace is a <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear mapping</a>. That is,<a href="#cite_note-:1-1">[1]</a><a href="#cite_note-:2-2">[2]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} )\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} )\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> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>c</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} )\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} )\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/872b65c362c45674fb790530e8f17a324544cfda" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.248ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} )\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} )\end{aligned}}}"> for all square matrices A and B, and all <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a> c.<a href="#cite_note-LipschutzLipson-3">[3]</a>: 34  A matrix and its <a href="/wiki/Transpose" title="Transpose">transpose</a> have the same trace:<a href="#cite_note-:1-1">[1]</a><a href="#cite_note-:2-2">[2]</a><a href="#cite_note-LipschutzLipson-3">[3]</a>: 34  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} )=\operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} )=\operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/726b465ce428ac298a9a1205058188417fac28a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.707ex; height:3.343ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} )=\operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\right).}"> This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal. <div class="mw-heading mw-heading3"><h3 id="Trace_of_a_product">Trace of a product</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=5" title="Edit section: Trace of a product">edit</a>]</div> The trace of a square matrix which is the product of two matrices can be rewritten as the sum of entry-wise products of their elements, i.e. as the sum of all elements of their <a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a>. Phrased directly, if A and B are two m × n matrices, then: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {B} \right)=\operatorname {tr} \left(\mathbf {A} \mathbf {B} ^{\mathsf {T}}\right)=\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {A} \right)=\operatorname {tr} \left(\mathbf {B} \mathbf {A} ^{\mathsf {T}}\right)=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ij}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {B} \right)=\operatorname {tr} \left(\mathbf {A} \mathbf {B} ^{\mathsf {T}}\right)=\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {A} \right)=\operatorname {tr} \left(\mathbf {B} \mathbf {A} ^{\mathsf {T}}\right)=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ij}\;.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042cb077b86b0c9aba11085d7aa27a3cd3018cdc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:63.224ex; height:7.176ex;" alt="{\displaystyle \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {B} \right)=\operatorname {tr} \left(\mathbf {A} \mathbf {B} ^{\mathsf {T}}\right)=\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {A} \right)=\operatorname {tr} \left(\mathbf {B} \mathbf {A} ^{\mathsf {T}}\right)=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ij}\;.}"> If one views any real m × n matrix as a vector of length mn (an operation called <a href="/wiki/Vectorization_(mathematics)" title="Vectorization (mathematics)">vectorization</a>) then the above operation on A and B coincides with the standard <a href="/wiki/Dot_product" title="Dot product">dot product</a>. According to the above expression, tr(A⊤A) is a sum of squares and hence is nonnegative, equal to zero if and only if A is zero.<a href="#cite_note-HornJohnson-4">[4]</a>: 7  Furthermore, as noted in the above formula, tr(A⊤B) = tr(B⊤A). These demonstrate the positive-definiteness and symmetry required of an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a>; it is common to call tr(A⊤B) the <a href="/wiki/Frobenius_inner_product" title="Frobenius inner product">Frobenius inner product</a> of A and B. This is a natural inner product on the <a href="/wiki/Vector_space" title="Vector space">vector space</a> of all real matrices of fixed dimensions. The <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> derived from this inner product is called the <a href="/wiki/Frobenius_norm" class="mw-redirect" title="Frobenius norm">Frobenius norm</a>, and it satisfies a submultiplicative property, as can be proven with the <a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \left[\operatorname {tr} (\mathbf {A} \mathbf {B} )\right]^{2}\leq \operatorname {tr} \left(\mathbf {A} ^{2}\right)\operatorname {tr} \left(\mathbf {B} ^{2}\right)\leq \left[\operatorname {tr} (\mathbf {A} )\right]^{2}\left[\operatorname {tr} (\mathbf {B} )\right]^{2}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msup> <mrow> <mo>[</mo> <mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>≤</mo> <msup> <mrow> <mo>[</mo> <mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>[</mo> <mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \left[\operatorname {tr} (\mathbf {A} \mathbf {B} )\right]^{2}\leq \operatorname {tr} \left(\mathbf {A} ^{2}\right)\operatorname {tr} \left(\mathbf {B} ^{2}\right)\leq \left[\operatorname {tr} (\mathbf {A} )\right]^{2}\left[\operatorname {tr} (\mathbf {B} )\right]^{2}\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560e619293d2e2483ce1d28f70a4ccfe5e8a1f50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:51.755ex; height:3.509ex;" alt="{\displaystyle 0\leq \left[\operatorname {tr} (\mathbf {A} \mathbf {B} )\right]^{2}\leq \operatorname {tr} \left(\mathbf {A} ^{2}\right)\operatorname {tr} \left(\mathbf {B} ^{2}\right)\leq \left[\operatorname {tr} (\mathbf {A} )\right]^{2}\left[\operatorname {tr} (\mathbf {B} )\right]^{2}\ ,}"> if A and B are real <a href="/wiki/Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">positive semi-definite matrices</a> of the same size. The Frobenius inner product and norm arise frequently in <a href="/wiki/Matrix_calculus" title="Matrix calculus">matrix calculus</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>. The Frobenius inner product may be extended to a <a href="/wiki/Hermitian_inner_product" class="mw-redirect" title="Hermitian inner product">hermitian inner product</a> on the <a href="/wiki/Complex_vector_space" class="mw-redirect" title="Complex vector space">complex vector space</a> of all complex matrices of a fixed size, by replacing B by its <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a>. The symmetry of the Frobenius inner product may be phrased more directly as follows: the matrices in the trace of a product can be switched without changing the result. If A and B are m × n and n × m real or complex matrices, respectively, then<a href="#cite_note-:1-1">[1]</a><a href="#cite_note-:2-2">[2]</a><a href="#cite_note-LipschutzLipson-3">[3]</a>: 34 <a href="#cite_note-5">[note 1]</a> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {B} \mathbf {A} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {B} \mathbf {A} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc15db0985db7141e60c24fffd5eea8f17eaf9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.191ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {B} \mathbf {A} )}"> </div> This is notable both for the fact that AB does not usually equal BA, and also since the trace of either does not usually equal tr(A)tr(B).<a href="#cite_note-6">[note 2]</a> The <a href="/wiki/Similarity_invariance" title="Similarity invariance">similarity-invariance</a> of the trace, meaning that tr(A) = tr(P−1AP) for any square matrix A and any invertible matrix P of the same dimensions, is a fundamental consequence. This is proved by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\mathbf {P} ^{-1}(\mathbf {A} \mathbf {P} )\right)=\operatorname {tr} \left((\mathbf {A} \mathbf {P} )\mathbf {P} ^{-1}\right)=\operatorname {tr} (\mathbf {A} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\mathbf {P} ^{-1}(\mathbf {A} \mathbf {P} )\right)=\operatorname {tr} \left((\mathbf {A} \mathbf {P} )\mathbf {P} ^{-1}\right)=\operatorname {tr} (\mathbf {A} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c98f8ad16e54243381d5f7c09cb3d6f524dddf97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.011ex; height:3.343ex;" alt="{\displaystyle \operatorname {tr} \left(\mathbf {P} ^{-1}(\mathbf {A} \mathbf {P} )\right)=\operatorname {tr} \left((\mathbf {A} \mathbf {P} )\mathbf {P} ^{-1}\right)=\operatorname {tr} (\mathbf {A} ).}"> Similarity invariance is the crucial property of the trace in order to discuss traces of <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformations</a> as below. Additionally, for real column vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \in \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31d9e403535f49451e6e98fefa153f4843465af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.037ex; height:2.343ex;" alt="{\displaystyle \mathbf {a} \in \mathbb {R} ^{n}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} \in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} \in \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9028033a55d0f5968de3761f26fdc7bd4a2018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.222ex; height:2.343ex;" alt="{\displaystyle \mathbf {b} \in \mathbb {R} ^{n}}">, the trace of the outer product is equivalent to the inner product: <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\mathbf {b} \mathbf {a} ^{\textsf {T}}\right)=\mathbf {a} ^{\textsf {T}}\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\mathbf {b} \mathbf {a} ^{\textsf {T}}\right)=\mathbf {a} ^{\textsf {T}}\mathbf {b} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce64fe8ff53757f059a619c6034febced9b6d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.317ex; height:3.343ex;" alt="{\displaystyle \operatorname {tr} \left(\mathbf {b} \mathbf {a} ^{\textsf {T}}\right)=\mathbf {a} ^{\textsf {T}}\mathbf {b} }"> </div> <div class="mw-heading mw-heading3"><h3 id="Cyclic_property">Cyclic property</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=6" title="Edit section: Cyclic property">edit</a>]</div> More generally, the trace is invariant under <a href="/wiki/Circular_shift" title="Circular shift">circular shifts</a>, that is, <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} \mathbf {D} )=\operatorname {tr} (\mathbf {B} \mathbf {C} \mathbf {D} \mathbf {A} )=\operatorname {tr} (\mathbf {C} \mathbf {D} \mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {D} \mathbf {A} \mathbf {B} \mathbf {C} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} \mathbf {D} )=\operatorname {tr} (\mathbf {B} \mathbf {C} \mathbf {D} \mathbf {A} )=\operatorname {tr} (\mathbf {C} \mathbf {D} \mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {D} \mathbf {A} \mathbf {B} \mathbf {C} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac613609f5d3e4512815665a631f2c9a8990324" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.05ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} \mathbf {D} )=\operatorname {tr} (\mathbf {B} \mathbf {C} \mathbf {D} \mathbf {A} )=\operatorname {tr} (\mathbf {C} \mathbf {D} \mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {D} \mathbf {A} \mathbf {B} \mathbf {C} ).}"> </div> This is known as the cyclic property. Arbitrary permutations are not allowed: in general, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )\neq \operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )\neq \operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a96389c6447846eb88ab2697f3d3b3224ab11f21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.7ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )\neq \operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ).}"> However, if products of three <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> matrices are considered, any permutation is allowed, since: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )=\operatorname {tr} \left(\left(\mathbf {A} \mathbf {B} \mathbf {C} \right)^{\mathsf {T}}\right)=\operatorname {tr} (\mathbf {C} \mathbf {B} \mathbf {A} )=\operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )=\operatorname {tr} \left(\left(\mathbf {A} \mathbf {B} \mathbf {C} \right)^{\mathsf {T}}\right)=\operatorname {tr} (\mathbf {C} \mathbf {B} \mathbf {A} )=\operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33442c22cc00cebcaed23a2afe4f36c0bc62a752" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.979ex; height:4.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )=\operatorname {tr} \left(\left(\mathbf {A} \mathbf {B} \mathbf {C} \right)^{\mathsf {T}}\right)=\operatorname {tr} (\mathbf {C} \mathbf {B} \mathbf {A} )=\operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ),}"> where the first equality is because the traces of a matrix and its transpose are equal. Note that this is not true in general for more than three factors. <div class="mw-heading mw-heading3"><h3 id="Trace_of_a_Kronecker_product">Trace of a Kronecker product</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=7" title="Edit section: Trace of a Kronecker product">edit</a>]</div> The trace of the <a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a> of two matrices is the product of their traces: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {tr} (\mathbf {A} )\operatorname {tr} (\mathbf {B} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {tr} (\mathbf {A} )\operatorname {tr} (\mathbf {B} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc86820a3ee20a40a9002c67982ea7f8d16c8172" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.691ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {tr} (\mathbf {A} )\operatorname {tr} (\mathbf {B} ).}"> <div class="mw-heading mw-heading3"><h3 id="Characterization_of_the_trace">Characterization of the trace</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=8" title="Edit section: Characterization of the trace">edit</a>]</div> The following three properties: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} ),\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} ),\\\operatorname {tr} (\mathbf {A} \mathbf {B} )&=\operatorname {tr} (\mathbf {B} \mathbf {A} ),\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> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>c</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} ),\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} ),\\\operatorname {tr} (\mathbf {A} \mathbf {B} )&=\operatorname {tr} (\mathbf {B} \mathbf {A} ),\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e30b932881e99d64bf99e8b16236b7d1c57ab3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:28.895ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} ),\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} ),\\\operatorname {tr} (\mathbf {A} \mathbf {B} )&=\operatorname {tr} (\mathbf {B} \mathbf {A} ),\end{aligned}}}"> characterize the trace <a href="/wiki/Up_to" title="Up to">up to</a> a scalar multiple in the following sense: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is a <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> on the space of square matrices that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(xy)=f(yx),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(xy)=f(yx),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8591f01420fcc5c143d5b562748edf196c32f09d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.891ex; height:2.843ex;" alt="{\displaystyle f(xy)=f(yx),}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e779faa4557258ea25e8101f307317fc771a1ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.816ex; height:2.009ex;" alt="{\displaystyle \operatorname {tr} }"> are proportional.<a href="#cite_note-7">[note 3]</a> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrices, imposing the normalization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {I} )=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {I} )=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0c436b515ec27c9c76c7db2adbe23ae9435d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.595ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {I} )=n}"> makes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> equal to the trace. <div class="mw-heading mw-heading3"><h3 id="Trace_as_the_sum_of_eigenvalues">Trace as the sum of eigenvalues</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=9" title="Edit section: Trace as the sum of eigenvalues">edit</a>]</div> Given any n × n matrix A, there is <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}\lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}\lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c258dcda985d2993f7edc2c6e8666b44e71587d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.641ex; height:6.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}\lambda _{i}}"> </div> where λ1, ..., λn are the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of A counted with multiplicity. This holds true even if A is a real matrix and some (or all) of the eigenvalues are complex numbers. This may be regarded as a consequence of the existence of the <a href="/wiki/Jordan_canonical_form" class="mw-redirect" title="Jordan canonical form">Jordan canonical form</a>, together with the similarity-invariance of the trace discussed above. <div class="mw-heading mw-heading3"><h3 id="Trace_of_commutator">Trace of commutator</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=10" title="Edit section: Trace of commutator">edit</a>]</div> When both A and B are n × n matrices, the trace of the (ring-theoretic) <a href="/wiki/Commutator" title="Commutator">commutator</a> of A and B vanishes: tr([A, B]) = 0, because tr(AB) = tr(BA) and tr is linear. One can state this as "the trace is a map of <a href="/wiki/Lie_algebras" class="mw-redirect" title="Lie algebras">Lie algebras</a> gln → k from operators to scalars", as the commutator of scalars is trivial (it is an <a href="/wiki/Abelian_Lie_algebra" class="mw-redirect" title="Abelian Lie algebra">Abelian Lie algebra</a>). In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices. Conversely, any square matrix with zero trace is a linear combination of the commutators of pairs of matrices.<a href="#cite_note-8">[note 4]</a> Moreover, any square matrix with zero trace is <a href="/wiki/Bounded_operator" title="Bounded operator">unitarily equivalent</a> to a square matrix with diagonal consisting of all zeros. <div class="mw-heading mw-heading3"><h3 id="Traces_of_special_kinds_of_matrices">Traces of special kinds of matrices</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=11" title="Edit section: Traces of special kinds of matrices">edit</a>]</div> <div><ul><li>The trace of the n × n <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> is the dimension of the space, namely n. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}\right)=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}\right)=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcdbaee4d9080f4187e965287aca5461a23717a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.351ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}\right)=n}"> This leads to <a href="/wiki/Dimension_(vector_space)#Trace" title="Dimension (vector space)">generalizations of dimension using trace</a>.</li><li>The trace of a <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrix</a> is real, because the elements on the diagonal are real.</li><li>The trace of a <a href="/wiki/Permutation_matrix" title="Permutation matrix">permutation matrix</a> is the number of <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a> of the corresponding permutation, because the diagonal term aii is 1 if the ith point is fixed and 0 otherwise.</li><li>The trace of a <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projection matrix</a> is the dimension of the target space. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {P} _{\mathbf {X} }&=\mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\\[3pt]\Longrightarrow \operatorname {tr} \left(\mathbf {P} _{\mathbf {X} }\right)&=\operatorname {rank} (\mathbf {X} ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⟹</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {P} _{\mathbf {X} }&=\mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\\[3pt]\Longrightarrow \operatorname {tr} \left(\mathbf {P} _{\mathbf {X} }\right)&=\operatorname {rank} (\mathbf {X} ).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70987d6a6a11aafe6e269c2cd255a9f0675cafc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.657ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {P} _{\mathbf {X} }&=\mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\\[3pt]\Longrightarrow \operatorname {tr} \left(\mathbf {P} _{\mathbf {X} }\right)&=\operatorname {rank} (\mathbf {X} ).\end{aligned}}}"> The matrix PX is idempotent.</li><li>More generally, the trace of any <a href="/wiki/Idempotent_matrix" title="Idempotent matrix">idempotent matrix</a>, i.e. one with A2 = A, equals its own <a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a>.</li><li>The trace of a <a href="/wiki/Nilpotent_matrix" title="Nilpotent matrix">nilpotent matrix</a> is zero. <div class="paragraphbreak" style="margin-top:0.5em"></div> When the characteristic of the base field is zero, the converse also holds: if tr(Ak) = 0 for all k, then A is nilpotent. <div class="paragraphbreak" style="margin-top:0.5em"></div> When the characteristic n > 0 is positive, the identity in n dimensions is a counterexample, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}^{k}\right)=\operatorname {tr} \left(\mathbf {I} _{n}\right)=n\equiv 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>n</mi> <mo>≡</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}^{k}\right)=\operatorname {tr} \left(\mathbf {I} _{n}\right)=n\equiv 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45fa06dbb314ac1da529e38817112b6d24e13dba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.889ex; height:3.343ex;" alt="{\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}^{k}\right)=\operatorname {tr} \left(\mathbf {I} _{n}\right)=n\equiv 0}">, but the identity is not nilpotent.</li></ul></div> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_the_characteristic_polynomial">Relationship to the characteristic polynomial</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=12" title="Edit section: Relationship to the characteristic polynomial">edit</a>]</div> The trace of an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c251a891b3fb73008b29c6b63034175b87398209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.159ex; height:2.676ex;" alt="{\displaystyle t^{n-1}}"> in the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a>, possibly changed of sign, according to the convention in the definition of the characteristic polynomial. <div class="mw-heading mw-heading2"><h2 id="Relationship_to_eigenvalues">Relationship to eigenvalues</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=13" title="Edit section: Relationship to eigenvalues">edit</a>]</div> If A is a linear operator represented by a square matrix with <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> entries and if λ1, ..., λn are the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of A (listed according to their <a href="/wiki/Algebraic_multiplicity" class="mw-redirect" title="Algebraic multiplicity">algebraic multiplicities</a>), then <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i}\lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i}\lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbbcdafce019e87efab7729879e964c2b3750bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.641ex; height:5.509ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i}\lambda _{i}}"> </div> This follows from the fact that A is always <a href="/wiki/Similar_matrix" class="mw-redirect" title="Similar matrix">similar</a> to its <a href="/wiki/Jordan_form" class="mw-redirect" title="Jordan form">Jordan form</a>, an upper <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular matrix</a> having λ1, ..., λn on the main diagonal. In contrast, the <a href="/wiki/Determinant" title="Determinant">determinant</a> of A is the product of its eigenvalues; that is, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\mathbf {A} )=\prod _{i}\lambda _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(\mathbf {A} )=\prod _{i}\lambda _{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3794767756daba1020b4681e368bbf324006a18a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.315ex; height:5.509ex;" alt="{\displaystyle \det(\mathbf {A} )=\prod _{i}\lambda _{i}.}"> Everything in the present section applies as well to any square matrix with coefficients in an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>. <div class="mw-heading mw-heading3"><h3 id="Derivative_relationships">Derivative relationships</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=14" title="Edit section: Derivative relationships">edit</a>]</div> If ΔA is a square matrix with small entries and I denotes the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>, then we have approximately <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\mathbf {I} +\mathbf {\Delta A} )\approx 1+\operatorname {tr} (\mathbf {\Delta A} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Δ</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>≈</mo> <mn>1</mn> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Δ</mi> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(\mathbf {I} +\mathbf {\Delta A} )\approx 1+\operatorname {tr} (\mathbf {\Delta A} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d9c07ff48ba8d0cfc22cfa4cec514e5cbe1a01" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.758ex; height:2.843ex;" alt="{\displaystyle \det(\mathbf {I} +\mathbf {\Delta A} )\approx 1+\operatorname {tr} (\mathbf {\Delta A} ).}"> Precisely this means that the trace is the <a href="/wiki/Derivative" title="Derivative">derivative</a> of the <a href="/wiki/Determinant" title="Determinant">determinant</a> function at the identity matrix. <a href="/wiki/Jacobi%27s_formula" title="Jacobi's formula">Jacobi's formula</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\det(\mathbf {A} )=\operatorname {tr} {\big (}\operatorname {adj} (\mathbf {A} )\cdot d\mathbf {A} {\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\det(\mathbf {A} )=\operatorname {tr} {\big (}\operatorname {adj} (\mathbf {A} )\cdot d\mathbf {A} {\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef61ad7a570664556c82a88a1b3d7a6da084b4c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.39ex; height:3.176ex;" alt="{\displaystyle d\det(\mathbf {A} )=\operatorname {tr} {\big (}\operatorname {adj} (\mathbf {A} )\cdot d\mathbf {A} {\big )}}"> is more general and describes the <a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">differential</a> of the determinant at an arbitrary square matrix, in terms of the trace and the <a href="/wiki/Adjugate_matrix" title="Adjugate matrix">adjugate</a> of the matrix. From this (or from the connection between the trace and the eigenvalues), one can derive a relation between the trace function, the <a href="/wiki/Matrix_exponential" title="Matrix exponential">matrix exponential</a> function, and the determinant:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\exp(\mathbf {A} ))=\exp(\operatorname {tr} (\mathbf {A} )).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(\exp(\mathbf {A} ))=\exp(\operatorname {tr} (\mathbf {A} )).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd18d9b3ac8161abfa6d132bc839dac6fd9539d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.172ex; height:2.843ex;" alt="{\displaystyle \det(\exp(\mathbf {A} ))=\exp(\operatorname {tr} (\mathbf {A} )).}"> A related characterization of the trace applies to linear <a href="/wiki/Vector_field" title="Vector field">vector fields</a>. Given a matrix A, define a vector field F on Rn by F(x) = Ax. The components of this vector field are linear functions (given by the rows of A). Its <a href="/wiki/Divergence" title="Divergence">divergence</a> div F is a constant function, whose value is equal to tr(A). By the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>, one can interpret this in terms of flows: if F(x) represents the velocity of a fluid at location x and U is a region in Rn, the <a href="/wiki/Flow_network" title="Flow network">net flow</a> of the fluid out of U is given by tr(A) · vol(U), where vol(U) is the <a href="/wiki/Volume" title="Volume">volume</a> of U. The trace is a linear operator, hence it commutes with the derivative: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\operatorname {tr} (\mathbf {X} )=\operatorname {tr} (d\mathbf {X} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\operatorname {tr} (\mathbf {X} )=\operatorname {tr} (d\mathbf {X} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aef55b5a2d8b7e8b19b2610adeade48b2426ffb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.854ex; height:2.843ex;" alt="{\displaystyle d\operatorname {tr} (\mathbf {X} )=\operatorname {tr} (d\mathbf {X} ).}"> <div class="mw-heading mw-heading2"><h2 id="Trace_of_a_linear_operator">Trace of a linear operator</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=15" title="Edit section: Trace of a linear operator">edit</a>]</div> In general, given some linear map f : V → V (where V is a finite-<a href="/wiki/Dimension_(linear_algebra)" class="mw-redirect" title="Dimension (linear algebra)">dimensional</a> <a href="/wiki/Vector_space" title="Vector space">vector space</a>), we can define the trace of this map by considering the trace of a <a href="/wiki/Representation_theory" title="Representation theory">matrix representation</a> of f, that is, choosing a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for V and describing f as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to <a href="/wiki/Matrix_similarity" title="Matrix similarity">similar matrices</a>, allowing for the possibility of a basis-independent definition for the trace of a linear map. Such a definition can be given using the <a href="/wiki/Natural_isomorphism" class="mw-redirect" title="Natural isomorphism">canonical isomorphism</a> between the space End(V) of linear maps on V and V ⊗ V*, where V* is the <a href="/wiki/Dual_space" title="Dual space">dual space</a> of V. Let v be in V and let g be in V*. Then the trace of the indecomposable element v ⊗ g is defined to be g(v); the trace of a general element is defined by linearity. The trace of a linear map f : V → V can then be defined as the trace, in the above sense, of the element of V ⊗ V* corresponding to f under the above mentioned canonical isomorphism. Using an explicit basis for V and the corresponding dual basis for V*, one can show that this gives the same definition of the trace as given above. <div class="mw-heading mw-heading2"><h2 id="Numerical_algorithms">Numerical algorithms</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=16" title="Edit section: Numerical algorithms">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Stochastic_estimator">Stochastic estimator</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=17" title="Edit section: Stochastic estimator">edit</a>]</div> The trace can be estimated unbiasedly by "Hutchinson's trick":<a href="#cite_note-9">[5]</a><blockquote>Given any matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\in \mathbb {R} ^{n\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\in \mathbb {R} ^{n\times n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9ad075e2a73039f5cac4dee0f234a0be0f164a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.437ex; height:2.343ex;" alt="{\displaystyle W\in \mathbb {R} ^{n\times n}}">, and any random <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3e3a8441aef5cdf3a0c60c74c688bbf75d2ba6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle u\in \mathbb {R} ^{n}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[uu^{T}]=I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mi>u</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[uu^{T}]=I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a3c98828325626aa932d00cb0d1d312d00904f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.388ex; height:3.176ex;" alt="{\displaystyle E[uu^{T}]=I}">, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[u^{T}Wu]=tr(W)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>W</mi> <mi>u</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[u^{T}Wu]=tr(W)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cc42305c8e1dd65ccb16b79c599e10d8ebdb0d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.784ex; height:3.176ex;" alt="{\displaystyle E[u^{T}Wu]=tr(W)}">. (Proof: expand the expectation directly.)</blockquote>Usually, the random vector is sampled from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(0,I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(0,I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb35ed5536ed6be56d780239548a0541f73376df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.241ex; height:2.843ex;" alt="{\displaystyle N(0,I)}"> (normal distribution) or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\pm n^{-1/2}\}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\pm n^{-1/2}\}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7286b293ce7dec7b678739e436e940449df2e513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.723ex; height:3.343ex;" alt="{\displaystyle \{\pm n^{-1/2}\}^{n}}"> (<a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher distribution</a>). More sophisticated stochastic estimators of trace have been developed.<a href="#cite_note-10">[6]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=18" title="Edit section: Applications">edit</a>]</div> If a 2 x 2 real matrix has zero trace, its square is a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a>. The trace of a 2 × 2 <a href="/wiki/Complex_matrix" class="mw-redirect" title="Complex matrix">complex matrix</a> is used to classify <a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a>. First, the matrix is normalized to make its <a href="/wiki/Determinant" title="Determinant">determinant</a> equal to one. Then, if the square of the trace is 4, the corresponding transformation is parabolic. If the square is in the interval [0,4), it is elliptic. Finally, if the square is greater than 4, the transformation is loxodromic. See <a href="/wiki/M%C3%B6bius_transformation#Classification" title="Möbius transformation">classification of Möbius transformations</a>. The trace is used to define <a href="/wiki/Character_(mathematics)" title="Character (mathematics)">characters</a> of <a href="/wiki/Group_representation" title="Group representation">group representations</a>. Two representations A, B : G → GL(V) of a group G are equivalent (up to change of basis on V) if tr(A(g)) = tr(B(g)) for all g ∈ G. The trace also plays a central role in the distribution of <a href="/wiki/Quadratic_form_(statistics)" title="Quadratic form (statistics)">quadratic forms</a>. <div class="mw-heading mw-heading2"><h2 id="Lie_algebra">Lie algebra</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=19" title="Edit section: Lie algebra">edit</a>]</div> The trace is a map of Lie algebras <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} :{\mathfrak {gl}}_{n}\to K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>:</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} :{\mathfrak {gl}}_{n}\to K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dbe14dfa3aac01665a117de3f5c80a8c8732c6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.475ex; height:2.676ex;" alt="{\displaystyle \operatorname {tr} :{\mathfrak {gl}}_{n}\to K}"> from the Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {gl}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {gl}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7074085f669b257cf21f2f443dd433888e2597b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.042ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {gl}}_{n}}"> of linear operators on an n-dimensional space (n × n matrices with entries in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">) to the Lie algebra K of scalars; as K is Abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} ([\mathbf {A} ,\mathbf {B} ])=0{\text{ for each }}\mathbf {A} ,\mathbf {B} \in {\mathfrak {gl}}_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for each </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} ([\mathbf {A} ,\mathbf {B} ])=0{\text{ for each }}\mathbf {A} ,\mathbf {B} \in {\mathfrak {gl}}_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c43376dc9fb9d15ae49fde0373b2259e6f1f98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.666ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} ([\mathbf {A} ,\mathbf {B} ])=0{\text{ for each }}\mathbf {A} ,\mathbf {B} \in {\mathfrak {gl}}_{n}.}"> The kernel of this map, a matrix whose trace is <a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">zero</a>, is often said to be <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>traceless or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">trace free, and these matrices form the <a href="/wiki/Simple_Lie_algebra" title="Simple Lie algebra">simple Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c75cbc0e014c41c8cd1e7a40afbadf87eb79de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.041ex; width:2.941ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {sl}}_{n}}">, which is the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the <a href="/wiki/Special_linear_Lie_algebra" title="Special linear Lie algebra">special linear Lie algebra</a> is the matrices which do not alter volume of infinitesimal sets. In fact, there is an internal <a href="/wiki/Direct_sum_of_Lie_algebras" class="mw-redirect" title="Direct sum of Lie algebras">direct sum</a> decomposition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊕</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d14307faac39541ec34a86251564f220468d567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.947ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K}"> of operators/matrices into traceless operators/matrices and scalars operators/matrices. The projection map onto scalar operators can be expressed in terms of the trace, concretely as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \mapsto {\frac {1}{n}}\operatorname {tr} (\mathbf {A} )\mathbf {I} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \mapsto {\frac {1}{n}}\operatorname {tr} (\mathbf {A} )\mathbf {I} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8440b8939dac781c4fdd6aadefae5312e7030096" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.557ex; height:5.176ex;" alt="{\displaystyle \mathbf {A} \mapsto {\frac {1}{n}}\operatorname {tr} (\mathbf {A} )\mathbf {I} .}"> Formally, one can compose the trace (the <a href="/wiki/Counit" class="mw-redirect" title="Counit">counit</a> map) with the unit map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\to {\mathfrak {gl}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\to {\mathfrak {gl}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f20c092b3efe5dca3128cc3b344514d4b8da9f82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.722ex; height:2.676ex;" alt="{\displaystyle K\to {\mathfrak {gl}}_{n}}"> of "inclusion of <a href="/wiki/Scalar_transformation" class="mw-redirect" title="Scalar transformation">scalars</a>" to obtain a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {gl}}_{n}\to {\mathfrak {gl}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {gl}}_{n}\to {\mathfrak {gl}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff6d138e9e8942165241712f5a7ca18f920bf59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.697ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {gl}}_{n}\to {\mathfrak {gl}}_{n}}"> mapping onto scalars, and multiplying by n. Dividing by n makes this a projection, yielding the formula above. In terms of <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequences</a>, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to {\mathfrak {sl}}_{n}\to {\mathfrak {gl}}_{n}{\overset {\operatorname {tr} }{\to }}K\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">→</mo> <mi>tr</mi> </mover> </mrow> <mi>K</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to {\mathfrak {sl}}_{n}\to {\mathfrak {gl}}_{n}{\overset {\operatorname {tr} }{\to }}K\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea71da653ae4fd1896e793aea7840b8ec9e0d1d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.499ex; height:4.009ex;" alt="{\displaystyle 0\to {\mathfrak {sl}}_{n}\to {\mathfrak {gl}}_{n}{\overset {\operatorname {tr} }{\to }}K\to 0}"> which is analogous to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\to \operatorname {SL} _{n}\to \operatorname {GL} _{n}{\overset {\det }{\to }}K^{*}\to 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo stretchy="false">→</mo> <msub> <mi>SL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">→</mo> <mo movablelimits="true" form="prefix">det</mo> </mover> </mrow> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\to \operatorname {SL} _{n}\to \operatorname {GL} _{n}{\overset {\det }{\to }}K^{*}\to 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67ecd34d1d8a7e6ed7e2d68cb850c5187874bb3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.486ex; height:4.009ex;" alt="{\displaystyle 1\to \operatorname {SL} _{n}\to \operatorname {GL} _{n}{\overset {\det }{\to }}K^{*}\to 1}"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{*}=K\setminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mi>K</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{*}=K\setminus \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba4ef0d2b8f4aeae8ea88713be6cffc4391acfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.995ex; height:2.843ex;" alt="{\displaystyle K^{*}=K\setminus \{0\}}">) for <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>. However, the trace splits naturally (via <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e10667bad240500f5044257143510127e03d69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.72ex; height:2.843ex;" alt="{\displaystyle 1/n}"> times scalars) so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊕</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d14307faac39541ec34a86251564f220468d567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.947ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K}">, but the splitting of the determinant would be as the nth root times scalars, and this does not in general define a function, so the determinant does not split and the general linear group does not decompose: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}\neq \operatorname {SL} _{n}\times K^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>SL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}\neq \operatorname {SL} _{n}\times K^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8081e03f9ffeff402ddf650ea7935d7f855b8422" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.548ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}\neq \operatorname {SL} _{n}\times K^{*}.}"> <div class="mw-heading mw-heading3"><h3 id="Bilinear_forms">Bilinear forms</h3>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=20" title="Edit section: Bilinear forms">edit</a>]</div> The <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a> (where X, Y are square matrices) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(\mathbf {X} ,\mathbf {Y} )=\operatorname {tr} (\operatorname {ad} (\mathbf {X} )\operatorname {ad} (\mathbf {Y} ))\quad {\text{where }}\operatorname {ad} (\mathbf {X} )\mathbf {Y} =[\mathbf {X} ,\mathbf {Y} ]=\mathbf {X} \mathbf {Y} -\mathbf {Y} \mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ad</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mi>ad</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where </mtext> </mrow> <mi>ad</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(\mathbf {X} ,\mathbf {Y} )=\operatorname {tr} (\operatorname {ad} (\mathbf {X} )\operatorname {ad} (\mathbf {Y} ))\quad {\text{where }}\operatorname {ad} (\mathbf {X} )\mathbf {Y} =[\mathbf {X} ,\mathbf {Y} ]=\mathbf {X} \mathbf {Y} -\mathbf {Y} \mathbf {X} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22bb409f81757ce372882865f643a6a0bea2f615" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.347ex; height:2.843ex;" alt="{\displaystyle B(\mathbf {X} ,\mathbf {Y} )=\operatorname {tr} (\operatorname {ad} (\mathbf {X} )\operatorname {ad} (\mathbf {Y} ))\quad {\text{where }}\operatorname {ad} (\mathbf {X} )\mathbf {Y} =[\mathbf {X} ,\mathbf {Y} ]=\mathbf {X} \mathbf {Y} -\mathbf {Y} \mathbf {X} }"> is called the <a href="/wiki/Killing_form" title="Killing form">Killing form</a>, which is used for the classification of Lie algebras. The trace defines a bilinear form: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {X} ,\mathbf {Y} )\mapsto \operatorname {tr} (\mathbf {X} \mathbf {Y} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {X} ,\mathbf {Y} )\mapsto \operatorname {tr} (\mathbf {X} \mathbf {Y} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505aa84c8b6f50fd7de8beb0af51e5fc9964f602" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.808ex; height:2.843ex;" alt="{\displaystyle (\mathbf {X} ,\mathbf {Y} )\mapsto \operatorname {tr} (\mathbf {X} \mathbf {Y} ).}"> The form is symmetric, non-degenerate<a href="#cite_note-11">[note 5]</a> and associative in the sense that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {X} [\mathbf {Y} ,\mathbf {Z} ])=\operatorname {tr} ([\mathbf {X} ,\mathbf {Y} ]\mathbf {Z} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {X} [\mathbf {Y} ,\mathbf {Z} ])=\operatorname {tr} ([\mathbf {X} ,\mathbf {Y} ]\mathbf {Z} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f62f8baf0101168b07e65fc9742f52d64531b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.998ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {X} [\mathbf {Y} ,\mathbf {Z} ])=\operatorname {tr} ([\mathbf {X} ,\mathbf {Y} ]\mathbf {Z} ).}"> For a complex simple Lie algebra (such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}}"> <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">l</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb4ad7d6d6ad9e9b642bee9982302dbfad673f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.041ex; width:1.722ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {sl}}}">n), every such bilinear form is proportional to each other; in particular, to the Killing form[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]. Two matrices X and Y are said to be trace orthogonal if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {X} \mathbf {Y} )=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {X} \mathbf {Y} )=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3aba7072acad8bd35c7cf54063f8ffa5b9b60a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.572ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {X} \mathbf {Y} )=0.}"> There is a generalization to a general representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\rho ,{\mathfrak {g}},V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\rho ,{\mathfrak {g}},V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f98391c9ece20bfeb52e7a7a3d7fc1653ba1a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.038ex; height:2.843ex;" alt="{\displaystyle (\rho ,{\mathfrak {g}},V)}"> of a Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> is a homomorphism of Lie algebras <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho :{\mathfrak {g}}\rightarrow {\text{End}}(V).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>End</mtext> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho :{\mathfrak {g}}\rightarrow {\text{End}}(V).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185ed4c95997afc69eb8aa201ea57439c2cc73bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.336ex; height:2.843ex;" alt="{\displaystyle \rho :{\mathfrak {g}}\rightarrow {\text{End}}(V).}"> The trace form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{tr}}_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>tr</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{tr}}_{V}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbf97618ae07cc7a93b24aaf206529e161f109d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.312ex; height:2.343ex;" alt="{\displaystyle {\text{tr}}_{V}}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{End}}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>End</mtext> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{End}}(V)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c11660eaacd443816e20d7b3e142b73dc5a52bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.764ex; height:2.843ex;" alt="{\displaystyle {\text{End}}(V)}"> is defined as above. The bilinear form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (\mathbf {X} ,\mathbf {Y} )={\text{tr}}_{V}(\rho (\mathbf {X} )\rho (\mathbf {Y} ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>tr</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">)</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (\mathbf {X} ,\mathbf {Y} )={\text{tr}}_{V}(\rho (\mathbf {X} )\rho (\mathbf {Y} ))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657abf0611b9df9928b23d45e0dcb6a6cc0a08ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.549ex; height:2.843ex;" alt="{\displaystyle \phi (\mathbf {X} ,\mathbf {Y} )={\text{tr}}_{V}(\rho (\mathbf {X} )\rho (\mathbf {Y} ))}"> is symmetric and invariant due to cyclicity. <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=21" title="Edit section: Generalizations">edit</a>]</div> The concept of trace of a matrix is generalized to the <a href="/wiki/Trace_class" title="Trace class">trace class</a> of <a href="/wiki/Compact_operator" title="Compact operator">compact operators</a> on <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>, and the analog of the <a href="/wiki/Frobenius_norm" class="mw-redirect" title="Frobenius norm">Frobenius norm</a> is called the <a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a> norm. If K is a trace-class operator, then for any <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{n})_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e_{n})_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe71683a1e22e9868c30ee5df485a2cc9c88dc0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.33ex; height:2.843ex;" alt="{\displaystyle (e_{n})_{n}}">, the trace is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (K)=\sum _{n}\left\langle e_{n},Ke_{n}\right\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>K</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (K)=\sum _{n}\left\langle e_{n},Ke_{n}\right\rangle ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d02472a093fc8bd8a9b1e1201488822d3c9428d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.079ex; height:5.509ex;" alt="{\displaystyle \operatorname {tr} (K)=\sum _{n}\left\langle e_{n},Ke_{n}\right\rangle ,}"> and is finite and independent of the orthonormal basis.<a href="#cite_note-12">[7]</a> The <a href="/wiki/Partial_trace" title="Partial trace">partial trace</a> is another generalization of the trace that is operator-valued. The trace of a linear operator Z which lives on a product space A ⊗ B is equal to the partial traces over A and B: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (Z)=\operatorname {tr} _{A}\left(\operatorname {tr} _{B}(Z)\right)=\operatorname {tr} _{B}\left(\operatorname {tr} _{A}(Z)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>tr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>tr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>tr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>tr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (Z)=\operatorname {tr} _{A}\left(\operatorname {tr} _{B}(Z)\right)=\operatorname {tr} _{B}\left(\operatorname {tr} _{A}(Z)\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c271df4b64917fecb83d6011b2ff5af05dbefebf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.902ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (Z)=\operatorname {tr} _{A}\left(\operatorname {tr} _{B}(Z)\right)=\operatorname {tr} _{B}\left(\operatorname {tr} _{A}(Z)\right).}"> For more properties and a generalization of the partial trace, see <a href="/wiki/Traced_monoidal_category" title="Traced monoidal category">traced monoidal categories</a>. If A is a general <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> over a field k, then a trace on A is often defined to be any map tr : A ↦ k which vanishes on commutators; tr([a,b]) = 0 for all a, b ∈ A. Such a trace is not uniquely defined; it can always at least be modified by multiplication by a nonzero scalar. A <a href="/wiki/Supertrace" title="Supertrace">supertrace</a> is the generalization of a trace to the setting of <a href="/wiki/Superalgebra" title="Superalgebra">superalgebras</a>. The operation of <a href="/wiki/Tensor_contraction" title="Tensor contraction">tensor contraction</a> generalizes the trace to arbitrary tensors. Gomme and Klein (2011) define a matrix trace operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {trm} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>trm</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {trm} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de4ce8f83fcd8f6b68f236d750552ea30c1bf00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.752ex; height:2.009ex;" alt="{\displaystyle \operatorname {trm} }"> that operates on <a href="/wiki/Block_matrix" title="Block matrix">block matrices</a> and use it to compute second-order perturbation solutions to dynamic economic models without the need for <a href="/wiki/Tensor_notation" class="mw-redirect" title="Tensor notation">tensor notation</a>.<a href="#cite_note-13">[8]</a> <div class="mw-heading mw-heading2"><h2 id="Traces_in_the_language_of_tensor_products">Traces in the language of tensor products</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=22" title="Edit section: Traces in the language of tensor products">edit</a>]</div> Given a vector space V, there is a natural bilinear map V × V∗ → F given by sending (v, φ) to the scalar φ(v). The <a href="/wiki/Tensor_product#Universal_property" title="Tensor product">universal property</a> of the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> V ⊗ V∗ automatically implies that this bilinear map is induced by a linear functional on V ⊗ V∗.<a href="#cite_note-kassel-14">[9]</a> Similarly, there is a natural bilinear map V × V∗ → Hom(V, V) given by sending (v, φ) to the linear map w ↦ φ(w)v. The universal property of the tensor product, just as used previously, says that this bilinear map is induced by a linear map V ⊗ V∗ → Hom(V, V). If V is finite-dimensional, then this linear map is a <a href="/wiki/Linear_isomorphism" class="mw-redirect" title="Linear isomorphism">linear isomorphism</a>.<a href="#cite_note-kassel-14">[9]</a> This fundamental fact is a straightforward consequence of the existence of a (finite) basis of V, and can also be phrased as saying that any linear map V → V can be written as the sum of (finitely many) rank-one linear maps. Composing the inverse of the isomorphism with the linear functional obtained above results in a linear functional on Hom(V, V). This linear functional is exactly the same as the trace. Using the definition of trace as the sum of diagonal elements, the matrix formula tr(AB) = tr(BA) is straightforward to prove, and was given above. In the present perspective, one is considering linear maps S and T, and viewing them as sums of rank-one maps, so that there are linear functionals φi and ψj and nonzero vectors vi and wj such that S(u) = Σφi(u)vi and T(u) = Σψj(u)wj for any u in V. Then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S\circ T)(u)=\sum _{i}\varphi _{i}\left(\sum _{j}\psi _{j}(u)w_{j}\right)v_{i}=\sum _{i}\sum _{j}\psi _{j}(u)\varphi _{i}(w_{j})v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>∘</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S\circ T)(u)=\sum _{i}\varphi _{i}\left(\sum _{j}\psi _{j}(u)w_{j}\right)v_{i}=\sum _{i}\sum _{j}\psi _{j}(u)\varphi _{i}(w_{j})v_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/047f0bc3f75041bfacdd02105d0b6eec0aa60dcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:62.474ex; height:7.676ex;" alt="{\displaystyle (S\circ T)(u)=\sum _{i}\varphi _{i}\left(\sum _{j}\psi _{j}(u)w_{j}\right)v_{i}=\sum _{i}\sum _{j}\psi _{j}(u)\varphi _{i}(w_{j})v_{i}}"></dd></dl> for any u in V. The rank-one linear map u ↦ ψj(u)φi(wj)vi has trace ψj(vi)φi(wj) and so <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (S\circ T)=\sum _{i}\sum _{j}\psi _{j}(v_{i})\varphi _{i}(w_{j})=\sum _{j}\sum _{i}\varphi _{i}(w_{j})\psi _{j}(v_{i}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>∘</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (S\circ T)=\sum _{i}\sum _{j}\psi _{j}(v_{i})\varphi _{i}(w_{j})=\sum _{j}\sum _{i}\varphi _{i}(w_{j})\psi _{j}(v_{i}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d493cda33a4dd3e22c44b22c18533c29c471104d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:56.493ex; height:5.843ex;" alt="{\displaystyle \operatorname {tr} (S\circ T)=\sum _{i}\sum _{j}\psi _{j}(v_{i})\varphi _{i}(w_{j})=\sum _{j}\sum _{i}\varphi _{i}(w_{j})\psi _{j}(v_{i}).}"></dd></dl> Following the same procedure with S and T reversed, one finds exactly the same formula, proving that tr(S ∘ T) equals tr(T ∘ S). The above proof can be regarded as being based upon tensor products, given that the fundamental identity of End(V) with V ⊗ V∗ is equivalent to the expressibility of any linear map as the sum of rank-one linear maps. As such, the proof may be written in the notation of tensor products. Then one may consider the multilinear map V × V∗ × V × V∗ → V ⊗ V∗ given by sending (v, φ, w, ψ) to φ(w)v ⊗ ψ. Further composition with the trace map then results in φ(w)ψ(v), and this is unchanged if one were to have started with (w, ψ, v, φ) instead. One may also consider the bilinear map End(V) × End(V) → End(V) given by sending (f, g) to the composition f ∘ g, which is then induced by a linear map End(V) ⊗ End(V) → End(V). It can be seen that this coincides with the linear map V ⊗ V∗ ⊗ V ⊗ V∗ → V ⊗ V∗. The established symmetry upon composition with the trace map then establishes the equality of the two traces.<a href="#cite_note-kassel-14">[9]</a> For any finite dimensional vector space V, there is a natural linear map F → V ⊗ V'; in the language of linear maps, it assigns to a scalar c the linear map c⋅idV. Sometimes this is called coevaluation map, and the trace V ⊗ V' → F is called evaluation map.<a href="#cite_note-kassel-14">[9]</a> These structures can be axiomatized to define <a href="/wiki/Categorical_trace" title="Categorical trace">categorical traces</a> in the abstract setting of <a href="/wiki/Category_theory" title="Category theory">category theory</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=23" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Scalar_curvature#Definition" title="Scalar curvature">Trace of a tensor with respect to a metric tensor</a></li> <li><a href="/wiki/Characteristic_function_(probability_theory)#Matrix-valued_random_variables" title="Characteristic function (probability theory)">Characteristic function</a></li> <li><a href="/wiki/Field_trace" title="Field trace">Field trace</a></li> <li><a href="/wiki/Golden%E2%80%93Thompson_inequality" title="Golden–Thompson inequality">Golden–Thompson inequality</a></li> <li><a href="/wiki/Singular_trace" title="Singular trace">Singular trace</a></li> <li><a href="/wiki/Specht%27s_theorem" title="Specht's theorem">Specht's theorem</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Trace_identity" title="Trace identity">Trace identity</a></li> <li><a href="/wiki/Trace_inequalities" class="mw-redirect" title="Trace inequalities">Trace inequalities</a></li> <li><a href="/wiki/Von_Neumann%27s_trace_inequality" class="mw-redirect" title="Von Neumann's trace inequality">von Neumann's trace inequality</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=24" title="Edit section: Notes">edit</a>]</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-5"><a href="#cite_ref-5">^</a> This is immediate from the definition of the <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\sum _{i=1}^{m}\left(\mathbf {A} \mathbf {B} \right)_{ii}=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\sum _{j=1}^{n}\sum _{i=1}^{m}b_{ji}a_{ij}=\sum _{j=1}^{n}\left(\mathbf {B} \mathbf {A} \right)_{jj}=\operatorname {tr} (\mathbf {B} \mathbf {A} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\sum _{i=1}^{m}\left(\mathbf {A} \mathbf {B} \right)_{ii}=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\sum _{j=1}^{n}\sum _{i=1}^{m}b_{ji}a_{ij}=\sum _{j=1}^{n}\left(\mathbf {B} \mathbf {A} \right)_{jj}=\operatorname {tr} (\mathbf {B} \mathbf {A} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9547654fa532867d3cc5cb0efa6ed235eccc54e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:78.461ex; height:7.176ex;" alt="{\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\sum _{i=1}^{m}\left(\mathbf {A} \mathbf {B} \right)_{ii}=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\sum _{j=1}^{n}\sum _{i=1}^{m}b_{ji}a_{ij}=\sum _{j=1}^{n}\left(\mathbf {B} \mathbf {A} \right)_{jj}=\operatorname {tr} (\mathbf {B} \mathbf {A} ).}"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> For example, if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}0&0\\1&0\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </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 \mathbf {A} ={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}0&0\\1&0\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec594d973a61e9e11068363a6f8809be3dc7658" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.761ex; height:6.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}0&0\\1&0\end{pmatrix}},}"> then the product is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {AB} ={\begin{pmatrix}1&0\\0&0\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {AB} ={\begin{pmatrix}1&0\\0&0\end{pmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d22f844ff810421b27f93859dd5633c0ca7a579c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.486ex; height:6.176ex;" alt="{\displaystyle \mathbf {AB} ={\begin{pmatrix}1&0\\0&0\end{pmatrix}},}"> and the traces are tr(AB) = 1 ≠ 0 ⋅ 0 = tr(A)tr(B). </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> Proof: Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b6a2274e22dc1d2778c28f3ce5b946d90ba2756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.561ex; height:2.343ex;" alt="{\displaystyle e_{ij}}"> the standard basis and note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(e_{ij}\right)=f\left(e_{i}e_{j}^{\top }\right)=f\left(e_{i}e_{1}^{\top }e_{1}e_{j}^{\top }\right)=f\left(e_{1}e_{j}^{\top }e_{i}e_{1}^{\top }\right)=f\left(0\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(e_{ij}\right)=f\left(e_{i}e_{j}^{\top }\right)=f\left(e_{i}e_{1}^{\top }e_{1}e_{j}^{\top }\right)=f\left(e_{1}e_{j}^{\top }e_{i}e_{1}^{\top }\right)=f\left(0\right)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97eed0157fcbe948681bf9f71d302288aff202dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.548ex; height:4.843ex;" alt="{\displaystyle f\left(e_{ij}\right)=f\left(e_{i}e_{j}^{\top }\right)=f\left(e_{i}e_{1}^{\top }e_{1}e_{j}^{\top }\right)=f\left(e_{1}e_{j}^{\top }e_{i}e_{1}^{\top }\right)=f\left(0\right)=0}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\neq j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\neq j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95aeb406bb427ac96806bc00c30c91d31b858be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.676ex;" alt="{\displaystyle i\neq j}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(e_{jj}\right)=f\left(e_{11}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(e_{jj}\right)=f\left(e_{11}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/950548eaad4e62d7da9be1e34916acffc94d8555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.679ex; height:3.009ex;" alt="{\displaystyle f\left(e_{jj}\right)=f\left(e_{11}\right)}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {A} )=\sum _{i,j}[\mathbf {A} ]_{ij}f\left(e_{ij}\right)=\sum _{i}[\mathbf {A} ]_{ii}f\left(e_{11}\right)=f\left(e_{11}\right)\operatorname {tr} (\mathbf {A} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>)</mo> </mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {A} )=\sum _{i,j}[\mathbf {A} ]_{ij}f\left(e_{ij}\right)=\sum _{i}[\mathbf {A} ]_{ii}f\left(e_{11}\right)=f\left(e_{11}\right)\operatorname {tr} (\mathbf {A} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39962edcfc3937240db861d2f696600925201688" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:56.167ex; height:5.843ex;" alt="{\displaystyle f(\mathbf {A} )=\sum _{i,j}[\mathbf {A} ]_{ij}f\left(e_{ij}\right)=\sum _{i}[\mathbf {A} ]_{ii}f\left(e_{11}\right)=f\left(e_{11}\right)\operatorname {tr} (\mathbf {A} ).}"> More abstractly, this corresponds to the decomposition <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⊕</mo> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c20ebe325f1d1ec9295dab7cd625daad32d1b20" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.739ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus k,}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e3cc6cb1fb477706acf1c248d6f66882df7176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.364ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)}"> (equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} ([A,B])=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} ([A,B])=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa090c8b1652e285edea85faddd3d357cf34b9d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.721ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} ([A,B])=0}">) defines the trace on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}_{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc728def492a16ea5866c86cb4a8a1a9e7320997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.041ex; width:3.588ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {sl}}_{n},}"> which has complement the scalar matrices, and leaves one degree of freedom: any such map is determined by its value on scalars, which is one scalar parameter and hence all are multiple of the trace, a nonzero such map. </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Proof: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c75cbc0e014c41c8cd1e7a40afbadf87eb79de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.041ex; width:2.941ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {sl}}_{n}}"> is a <a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">semisimple Lie algebra</a> and thus every element in it is a linear combination of commutators of some pairs of elements, otherwise the <a href="/wiki/Derived_algebra" class="mw-redirect" title="Derived algebra">derived algebra</a> would be a proper ideal. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> This follows from the fact that tr(A*A) = 0 <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> A = 0. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=25" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-:1-1">^ <a href="#cite_ref-:1_1-0">a</a> <a href="#cite_ref-:1_1-1">b</a> <a href="#cite_ref-:1_1-2">c</a> <a href="#cite_ref-:1_1-3">d</a> <a href="#cite_ref-:1_1-4">e</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html">"Rank, trace, determinant, transpose, and inverse of matrices"</a>. fourier.eng.hmc.edu. Retrieved 2020-09-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=fourier.eng.hmc.edu&rft.atitle=Rank%2C+trace%2C+determinant%2C+transpose%2C+and+inverse+of+matrices&rft_id=http%3A%2F%2Ffourier.eng.hmc.edu%2Fe161%2Flectures%2Falgebra%2Fnode2.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrace+%28linear+algebra%29" class="Z3988"> </li> <li id="cite_note-:2-2">^ <a href="#cite_ref-:2_2-0">a</a> <a href="#cite_ref-:2_2-1">b</a> <a href="#cite_ref-:2_2-2">c</a> <a href="#cite_ref-:2_2-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein2003" class="citation encyclopaedia cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> (2003) [1999]. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/MatrixTrace.html">"Trace (matrix)"</a>. In Weisstein, Eric W. (ed.). <a href="/wiki/CRC_Concise_Encyclopedia_of_Mathematics" title="CRC Concise Encyclopedia of Mathematics">CRC Concise Encyclopedia of Mathematics</a> (2nd ed.). Boca Raton, FL: <a href="/wiki/Chapman_%26_Hall" title="Chapman & Hall">Chapman & Hall</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9781420035223">10.1201/9781420035223</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58488-347-2" title="Special:BookSources/1-58488-347-2"><bdi>1-58488-347-2</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1944431">1944431</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1079.00009">1079.00009</a>. Retrieved 2020-09-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Trace+%28matrix%29&rft.btitle=CRC+Concise+Encyclopedia+of+Mathematics&rft.place=Boca+Raton%2C+FL&rft.edition=2nd&rft.pub=Chapman+%26+Hall&rft.date=2003&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1079.00009%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1944431%23id-name%3DMR&rft_id=info%3Adoi%2F10.1201%2F9781420035223&rft.isbn=1-58488-347-2&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMatrixTrace.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrace+%28linear+algebra%29" class="Z3988"> </li> <li id="cite_note-LipschutzLipson-3">^ <a href="#cite_ref-LipschutzLipson_3-0">a</a> <a href="#cite_ref-LipschutzLipson_3-1">b</a> <a href="#cite_ref-LipschutzLipson_3-2">c</a> <a href="#cite_ref-LipschutzLipson_3-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLipschutzLipson2005" class="citation book cs1">Lipschutz, Seymour; Lipson, Marc (September 2005). Theory and Problems of Linear Algebra. Schaum's Outline. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780070605022" title="Special:BookSources/9780070605022"><bdi>9780070605022</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+and+Problems+of+Linear+Algebra&rft.series=Schaum%27s+Outline&rft.pub=McGraw-Hill&rft.date=2005-09&rft.isbn=9780070605022&rft.aulast=Lipschutz&rft.aufirst=Seymour&rft.au=Lipson%2C+Marc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrace+%28linear+algebra%29" class="Z3988"> </li> <li id="cite_note-HornJohnson-4"><a href="#cite_ref-HornJohnson_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornJohnson2013" class="citation book cs1">Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (2nd ed.). 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Communications in Statistics - Simulation and Computation. 18 (3): 1059–1076. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610918908812806">10.1080/03610918908812806</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/0361-0918">0361-0918</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+in+Statistics+-+Simulation+and+Computation&rft.atitle=A+Stochastic+Estimator+of+the+Trace+of+the+Influence+Matrix+for+Laplacian+Smoothing+Splines&rft.volume=18&rft.issue=3&rft.pages=1059-1076&rft.date=1989-01&rft_id=info%3Adoi%2F10.1080%2F03610918908812806&rft.issn=0361-0918&rft.aulast=Hutchinson&rft.aufirst=M.F.&rft_id=http%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F03610918908812806&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrace+%28linear+algebra%29" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAvronToledo2011" class="citation journal cs1">Avron, Haim; Toledo, Sivan (2011-04-11). <a rel="nofollow" class="external text" href="https://doi.org/10.1145/1944345.1944349">"Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix"</a>. 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Klein (2011). "Second-order approximation of dynamic models without the use of tensors". Journal of Economic Dynamics & Control. 35 (4): 604–615. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jedc.2010.10.006">10.1016/j.jedc.2010.10.006</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Economic+Dynamics+%26+Control&rft.atitle=Second-order+approximation+of+dynamic+models+without+the+use+of+tensors&rft.volume=35&rft.issue=4&rft.pages=604-615&rft.date=2011&rft_id=info%3Adoi%2F10.1016%2Fj.jedc.2010.10.006&rft.au=P.+Gomme%2C+P.+Klein&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrace+%28linear+algebra%29" class="Z3988"> </li> <li id="cite_note-kassel-14">^ <a href="#cite_ref-kassel_14-0">a</a> <a href="#cite_ref-kassel_14-1">b</a> <a href="#cite_ref-kassel_14-2">c</a> <a href="#cite_ref-kassel_14-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKassel1995" class="citation book cs1">Kassel, Christian (1995). 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The Theory of Matrices. Translated by <a href="/wiki/Kurt_Hirsch" title="Kurt Hirsch">Hirsch, K.A.</a> New York, NY: <a href="/wiki/Chelsea_Publishing_Company" title="Chelsea Publishing Company">Chelsea Publishing Company</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0107649">0107649</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Matrices&rft.place=New+York%2C+NY&rft.pub=Chelsea+Publishing+Company&rft.date=1959&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0107649%23id-name%3DMR&rft.aulast=Gantmacher&rft.aufirst=F.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrace+%28linear+algebra%29" class="Z3988"></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornJohnson2013" class="citation book cs1"><a href="/wiki/Roger_Horn" title="Roger Horn">Horn, R.A.</a>; <a href="/wiki/Charles_Royal_Johnson" title="Charles Royal Johnson">Johnson, C.R.</a> (2013) [1985]. Matrix Analysis (2nd ed.). Cambridge, UK: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-54823-6" title="Special:BookSources/978-0-521-54823-6"><bdi>978-0-521-54823-6</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2978290">2978290</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis&rft.place=Cambridge%2C+UK&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-0-521-54823-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2978290%23id-name%3DMR&rft.aulast=Horn&rft.aufirst=R.A.&rft.au=Johnson%2C+C.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrace+%28linear+algebra%29" class="Z3988"></li></ul> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang2004" class="citation book cs1"><a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Strang, G.</a> (2004) [1976]. Linear Algebra and its Applications (4th ed.). <a href="/wiki/Cengage_Learning" class="mw-redirect" title="Cengage Learning">Cengage Learning</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-003010567-8" title="Special:BookSources/978-003010567-8"><bdi>978-003010567-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+its+Applications&rft.edition=4th&rft.pub=Cengage+Learning&rft.date=2004&rft.isbn=978-003010567-8&rft.aulast=Strang&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrace+%28linear+algebra%29" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Trace_(linear_algebra)&action=edit&section=26" title="Edit section: External links">edit</a>]</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=Trace_of_a_square_matrix">"Trace of a square matrix"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Trace+of+a+square+matrix&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DTrace_of_a_square_matrix&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATrace+%28linear+algebra%29" class="Z3988"></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Trace_(linear_algebra)&oldid=1256943232">https://en.wikipedia.org/w/index.php?title=Trace_(linear_algebra)&oldid=1256943232</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Linear algebra</a></li><li><a href="/wiki/Category:Matrix_theory" title="Category:Matrix theory">Matrix theory</a></li><li><a href="/wiki/Category:Trace_theory" title="Category:Trace theory">Trace theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_November_2023" title="Category:Articles needing additional references from November 2023">Articles needing additional references from November 2023</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_June_2022" title="Category:Articles with unsourced statements from June 2022">Articles with unsourced statements from June 2022</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 12 November 2024, at 10:06 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Trace (linear algebra) - Wikipedia