Autovettore e autovalore

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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Introduzione_informale" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Introduzione_informale"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Introduzione informale</span> </div> </a> <ul id="toc-Introduzione_informale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definizione" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definizione"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definizione</span> </div> </a> <button aria-controls="toc-Definizione-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Definizione</span> </button> <ul id="toc-Definizione-sublist" class="vector-toc-list"> <li id="toc-Descrizione_matriciale_e_autovettore_sinistro" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Descrizione_matriciale_e_autovettore_sinistro"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Descrizione matriciale e autovettore sinistro</span> </div> </a> <ul id="toc-Descrizione_matriciale_e_autovettore_sinistro-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Autofunzioni" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Autofunzioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Autofunzioni</span> </div> </a> <ul id="toc-Autofunzioni-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Polinomio_caratteristico" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polinomio_caratteristico"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Polinomio caratteristico</span> </div> </a> <ul id="toc-Polinomio_caratteristico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diagonalizzabilità" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diagonalizzabilità"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Diagonalizzabilità</span> </div> </a> <button aria-controls="toc-Diagonalizzabilità-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Diagonalizzabilità</span> </button> <ul id="toc-Diagonalizzabilità-sublist" class="vector-toc-list"> <li id="toc-Il_teorema_spettrale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Il_teorema_spettrale"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Il teorema spettrale</span> </div> </a> <ul id="toc-Il_teorema_spettrale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Spettro_di_un_operatore" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spettro_di_un_operatore"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Spettro di un operatore</span> </div> </a> <button aria-controls="toc-Spettro_di_un_operatore-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Spettro di un operatore</span> </button> <ul id="toc-Spettro_di_un_operatore-sublist" class="vector-toc-list"> <li id="toc-Operatori_aggiunti_e_autoaggiunti" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operatori_aggiunti_e_autoaggiunti"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Operatori aggiunti e autoaggiunti</span> </div> </a> <ul id="toc-Operatori_aggiunti_e_autoaggiunti-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applicazioni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applicazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applicazioni</span> </div> </a> <button aria-controls="toc-Applicazioni-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Applicazioni</span> </button> <ul id="toc-Applicazioni-sublist" class="vector-toc-list"> <li id="toc-Operatori_in_meccanica_quantistica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operatori_in_meccanica_quantistica"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Operatori in meccanica quantistica</span> </div> </a> <ul id="toc-Operatori_in_meccanica_quantistica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teoria_dei_numeri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teoria_dei_numeri"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Teoria dei numeri</span> </div> </a> <ul id="toc-Teoria_dei_numeri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Autofacce" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Autofacce"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Autofacce</span> </div> </a> <ul id="toc-Autofacce-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensore_d'inerzia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensore_d'inerzia"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Tensore d'inerzia</span> </div> </a> <ul id="toc-Tensore_d'inerzia-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Esempi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Esempi"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Esempi</span> </div> </a> <button aria-controls="toc-Esempi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Esempi</span> </button> <ul id="toc-Esempi-sublist" class="vector-toc-list"> <li id="toc-Esempi_nel_piano" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Esempi_nel_piano"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Esempi nel piano</span> </div> </a> <ul id="toc-Esempi_nel_piano-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Esempi_nello_spazio" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Esempi_nello_spazio"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Esempi nello spazio</span> </div> </a> <ul id="toc-Esempi_nello_spazio-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Esempi_di_calcolo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Esempi_di_calcolo"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Esempi di calcolo</span> </div> </a> <ul id="toc-Esempi_di_calcolo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostra/Nascondi l'indice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Autovettore e autovalore</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vai a una voce in un'altra lingua. Disponibile in 49 lingue" > <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-49" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">49 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D9%8A%D9%85_%D8%B0%D8%A7%D8%AA%D9%8A%D8%A9_%D9%88%D9%85%D8%AA%D8%AC%D9%87%D8%A7%D8%AA_%D8%B0%D8%A7%D8%AA%D9%8A%D8%A9" title="قيم ذاتية ومتجهات ذاتية - arabo" lang="ar" hreflang="ar" data-title="قيم ذاتية ومتجهات ذاتية" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Vector_propiu_y_valor_propiu" title="Vector propiu y valor propiu - asturiano" lang="ast" hreflang="ast" data-title="Vector propiu y valor propiu" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A3%D0%BB%D0%B0%D1%81%D0%BD%D1%8B%D1%8F_%D0%B2%D0%B5%D0%BA%D1%82%D0%B0%D1%80%D1%8B_%D1%96_%D1%9E%D0%BB%D0%B0%D1%81%D0%BD%D1%8B%D1%8F_%D0%B7%D0%BD%D0%B0%D1%87%D1%8D%D0%BD%D0%BD%D1%96" title="Уласныя вектары і ўласныя значэнні - bielorusso" lang="be" hreflang="be" data-title="Уласныя вектары і ўласныя значэнні" data-language-autonym="Беларуская" data-language-local-name="bielorusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A3%D0%BB%D0%B0%D1%81%D0%BD%D1%8B%D1%8F_%D0%BB%D1%96%D0%BA%D1%96,_%D0%B2%D1%8D%D0%BA%D1%82%D0%B0%D1%80%D1%8B_%D1%96_%D0%BF%D1%80%D0%B0%D1%81%D1%82%D0%BE%D1%80%D1%8B" title="Уласныя лікі, вэктары і прасторы - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Уласныя лікі, вэктары і прасторы" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BE%D0%B1%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%B8_%D1%81%D1%82%D0%BE%D0%B9%D0%BD%D0%BE%D1%81%D1%82%D0%B8_%D0%B8_%D1%81%D0%BE%D0%B1%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%B8_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8" title="Собствени стойности и собствени вектори - bulgaro" lang="bg" hreflang="bg" data-title="Собствени стойности и собствени вектори" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Valor_propi,_vector_propi_i_espai_propi" title="Valor propi, vector propi i espai propi - catalano" lang="ca" hreflang="ca" data-title="Valor propi, vector propi i espai propi" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vlastn%C3%AD_vektory_a_vlastn%C3%AD_%C4%8D%C3%ADsla" title="Vlastní vektory a vlastní čísla - ceco" lang="cs" hreflang="cs" data-title="Vlastní vektory a vlastní čísla" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Egenv%C3%A6rdi,_egenvektor_og_egenrum" title="Egenværdi, egenvektor og egenrum - danese" lang="da" hreflang="da" data-title="Egenværdi, egenvektor og egenrum" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Eigenwerte_und_Eigenvektoren" title="Eigenwerte und Eigenvektoren - tedesco" lang="de" hreflang="de" data-title="Eigenwerte und Eigenvektoren" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%99%CE%B4%CE%B9%CE%BF%CF%84%CE%B9%CE%BC%CE%AD%CF%82_%CE%BA%CE%B1%CE%B9_%CE%B9%CE%B4%CE%B9%CE%BF%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%B1" title="Ιδιοτιμές και ιδιοδιανύσματα - greco" lang="el" hreflang="el" data-title="Ιδιοτιμές και ιδιοδιανύσματα" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors - inglese" lang="en" hreflang="en" data-title="Eigenvalues and eigenvectors" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ejgeno_kaj_ejgenvektoro" title="Ejgeno kaj ejgenvektoro - esperanto" lang="eo" hreflang="eo" data-title="Ejgeno kaj ejgenvektoro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Vector,_valor_y_espacio_propios" title="Vector, valor y espacio propios - spagnolo" lang="es" hreflang="es" data-title="Vector, valor y espacio propios" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Omav%C3%A4%C3%A4rtus_ja_omavektor" title="Omaväärtus ja omavektor - estone" lang="et" hreflang="et" data-title="Omaväärtus ja omavektor" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Autobalioak_eta_autobektoreak" title="Autobalioak eta autobektoreak - basco" lang="eu" hreflang="eu" data-title="Autobalioak eta autobektoreak" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%82%D8%AF%D8%A7%D8%B1%D9%88%DB%8C%DA%98%D9%87_%D9%88_%D8%A8%D8%B1%D8%AF%D8%A7%D8%B1%D9%88%DB%8C%DA%98%D9%87" title="مقدارویژه و بردارویژه - persiano" lang="fa" hreflang="fa" data-title="مقدارویژه و بردارویژه" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ominaisarvo,_ominaisvektori_ja_ominaisavaruus" title="Ominaisarvo, ominaisvektori ja ominaisavaruus - finlandese" lang="fi" hreflang="fi" data-title="Ominaisarvo, ominaisvektori ja ominaisavaruus" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Valeur_propre,_vecteur_propre_et_espace_propre" title="Valeur propre, vecteur propre et espace propre - francese" lang="fr" hreflang="fr" data-title="Valeur propre, vecteur propre et espace propre" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Valor_propio,_vector_propio_e_espazo_propio" title="Valor propio, vector propio e espazo propio - galiziano" lang="gl" hreflang="gl" data-title="Valor propio, vector propio e espazo propio" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%AD%E0%A4%BF%E0%A4%B2%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%A3%E0%A4%BF%E0%A4%95_%E0%A4%AE%E0%A4%BE%E0%A4%A8_%E0%A4%A4%E0%A4%A5%E0%A4%BE_%E0%A4%85%E0%A4%AD%E0%A4%BF%E0%A4%B2%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%A3%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6" title="अभिलक्षणिक मान तथा अभिलक्षणिक सदिश - hindi" lang="hi" hreflang="hi" data-title="अभिलक्षणिक मान तथा अभिलक्षणिक सदिश" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Svojstvene_vrijednosti_i_svojstveni_vektori" title="Svojstvene vrijednosti i svojstveni vektori - croato" lang="hr" hreflang="hr" data-title="Svojstvene vrijednosti i svojstveni vektori" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Saj%C3%A1tvektor_%C3%A9s_saj%C3%A1t%C3%A9rt%C3%A9k" title="Sajátvektor és sajátérték - ungherese" lang="hu" hreflang="hu" data-title="Sajátvektor és sajátérték" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Eigenvalores_e_eigenvectores" title="Eigenvalores e eigenvectores - interlingua" lang="ia" hreflang="ia" data-title="Eigenvalores e eigenvectores" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Nilai_dan_vektor_eigen" title="Nilai dan vektor eigen - indonesiano" lang="id" hreflang="id" data-title="Nilai dan vektor eigen" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Eigen_gildi" title="Eigen gildi - islandese" lang="is" hreflang="is" data-title="Eigen gildi" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%BA%E6%9C%89%E5%80%A4%E3%81%A8%E5%9B%BA%E6%9C%89%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB" title="固有値と固有ベクトル - giapponese" lang="ja" hreflang="ja" data-title="固有値と固有ベクトル" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%A0%EC%9C%B3%EA%B0%92%EA%B3%BC_%EA%B3%A0%EC%9C%A0_%EB%B2%A1%ED%84%B0" title="고윳값과 고유 벡터 - coreano" lang="ko" hreflang="ko" data-title="고윳값과 고유 벡터" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tikrini%C5%B3_ver%C4%8Di%C5%B3_lygtis" title="Tikrinių verčių lygtis - lituano" lang="lt" hreflang="lt" data-title="Tikrinių verčių lygtis" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%AApa%C5%A1v%C4%93rt%C4%ABbas_un_%C4%ABpa%C5%A1vektori" title="Īpašvērtības un īpašvektori - lettone" lang="lv" hreflang="lv" data-title="Īpašvērtības un īpašvektori" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Eigenverdi,_eigenvektor_og_eigerom" title="Eigenverdi, eigenvektor og eigerom - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Eigenverdi, eigenvektor og eigerom" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%86%E0%A8%88%E0%A8%97%E0%A8%A8-%E0%A8%AE%E0%A9%81%E0%A9%B1%E0%A8%B2_%E0%A8%85%E0%A8%A4%E0%A9%87_%E0%A8%86%E0%A8%88%E0%A8%97%E0%A8%A8-%E0%A8%B5%E0%A9%88%E0%A8%95%E0%A8%9F%E0%A8%B0" title="ਆਈਗਨ-ਮੁੱਲ ਅਤੇ ਆਈਗਨ-ਵੈਕਟਰ - punjabi" lang="pa" hreflang="pa" data-title="ਆਈਗਨ-ਮੁੱਲ ਅਤੇ ਆਈਗਨ-ਵੈਕਟਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wektory_i_warto%C5%9Bci_w%C5%82asne" title="Wektory i wartości własne - polacco" lang="pl" hreflang="pl" data-title="Wektory i wartości własne" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%88%DB%8C%DA%98%DB%81_%D9%82%D8%AF%D8%B1" title="ویژہ قدر - Western Punjabi" lang="pnb" hreflang="pnb" data-title="ویژہ قدر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Autovalores_e_autovetores" title="Autovalores e autovetores - portoghese" lang="pt" hreflang="pt" data-title="Autovalores e autovetores" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Vectori_%C8%99i_valori_proprii" title="Vectori și valori proprii - rumeno" lang="ro" hreflang="ro" data-title="Vectori și valori proprii" data-language-autonym="Română" data-language-local-name="rumeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Svojstvene_vrijednosti_i_svojstveni_vektori" title="Svojstvene vrijednosti i svojstveni vektori - serbo-croato" lang="sh" hreflang="sh" data-title="Svojstvene vrijednosti i svojstveni vektori" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors - Simple English" lang="en-simple" hreflang="en-simple" data-title="Eigenvalues and eigenvectors" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Lastna_vrednost" title="Lastna vrednost - sloveno" lang="sl" hreflang="sl" data-title="Lastna vrednost" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Autovlerat_dhe_autovektor%C3%ABt" title="Autovlerat dhe autovektorët - albanese" lang="sq" hreflang="sq" data-title="Autovlerat dhe autovektorët" data-language-autonym="Shqip" data-language-local-name="albanese" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Sopstvene_vrednosti_i_sopstveni_vektori" title="Sopstvene vrednosti i sopstveni vektori - serbo" lang="sr" hreflang="sr" data-title="Sopstvene vrednosti i sopstveni vektori" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Egenv%C3%A4rde,_egenvektor_och_egenrum" title="Egenvärde, egenvektor och egenrum - svedese" lang="sv" hreflang="sv" data-title="Egenvärde, egenvektor och egenrum" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%90%E0%AE%95%E0%AF%86%E0%AE%A9%E0%AF%8D_%E0%AE%AE%E0%AE%A4%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="ஐகென் மதிப்பு - tamil" lang="ta" hreflang="ta" data-title="ஐகென் மதிப்பு" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%A9%E0%B8%93%E0%B8%B0%E0%B9%80%E0%B8%89%E0%B8%9E%E0%B8%B2%E0%B8%B0%E0%B9%81%E0%B8%A5%E0%B8%B0%E0%B9%80%E0%B8%A7%E0%B8%81%E0%B9%80%E0%B8%95%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%A9%E0%B8%93%E0%B8%B0%E0%B9%80%E0%B8%89%E0%B8%9E%E0%B8%B2%E0%B8%B0" title="ค่าลักษณะเฉพาะและเวกเตอร์ลักษณะเฉพาะ - thailandese" lang="th" hreflang="th" data-title="ค่าลักษณะเฉพาะและเวกเตอร์ลักษณะเฉพาะ" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%96zde%C4%9Fer,_%C3%B6zvekt%C3%B6r,_%C3%B6zuzay" title="Özdeğer, özvektör, özuzay - turco" lang="tr" hreflang="tr" data-title="Özdeğer, özvektör, özuzay" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%BB%D0%B0%D1%81%D0%BD%D1%96_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8_%D1%82%D0%B0_%D0%B2%D0%BB%D0%B0%D1%81%D0%BD%D1%96_%D0%B7%D0%BD%D0%B0%D1%87%D0%B5%D0%BD%D0%BD%D1%8F" title="Власні вектори та власні значення - ucraino" lang="uk" hreflang="uk" data-title="Власні вектори та власні значення" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%88%DB%8C%DA%98%DB%81_%D9%82%D8%AF%D8%B1" title="ویژہ قدر - urdu" lang="ur" hreflang="ur" data-title="ویژہ قدر" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gi%C3%A1_tr%E1%BB%8B_ri%C3%AAng_v%C3%A0_vect%C6%A1_ri%C3%AAng" title="Giá trị riêng và vectơ riêng - vietnamita" lang="vi" hreflang="vi" data-title="Giá trị riêng và vectơ riêng" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%89%B9%E5%BE%81%E5%80%BC%E5%92%8C%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F" title="特征值和特征向量 - cinese" lang="zh" hreflang="zh" data-title="特征值和特征向量" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%89%B9%E5%BE%B5%E5%80%BC%E3%80%81%E7%89%B9%E5%BE%B5%E5%90%91%E9%87%8F%E5%90%8C%E7%89%B9%E5%BE%B5%E7%A9%BA%E9%96%93" title="特徵值、特徵向量同特徵空間 - cantonese" lang="yue" hreflang="yue" data-title="特徵值、特徵向量同特徵空間" data-language-autonym="粵語" data-language-local-name="cantonese" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q190524#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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</nav> </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">Da Wikipedia, l'enciclopedia libera.</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="it" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mona_Lisa_with_eigenvector.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Mona_Lisa_with_eigenvector.png/260px-Mona_Lisa_with_eigenvector.png" decoding="async" width="260" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Mona_Lisa_with_eigenvector.png/390px-Mona_Lisa_with_eigenvector.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/25/Mona_Lisa_with_eigenvector.png/520px-Mona_Lisa_with_eigenvector.png 2x" data-file-width="1643" data-file-height="1395" /></a><figcaption>In questa trasformazione lineare della <a href="/wiki/Gioconda" title="Gioconda">Gioconda</a> l'immagine è modificata ma l'asse centrale verticale rimane fisso. Il vettore blu ha cambiato lievemente direzione, mentre quello rosso no. Quindi il vettore rosso è un autovettore della trasformazione e quello blu no. Inoltre, poiché il vettore rosso non è stato né allungato, né compresso, né ribaltato, il suo autovalore è 1. Tutti i vettori sull'asse verticale sono multipli scalari del vettore rosso, e sono tutti autovettori: assieme all'origine, formano l'autospazio relativo all'autovalore 1.</figcaption></figure> <p>In <a href="/wiki/Matematica" title="Matematica">matematica</a>, in particolare in <a href="/wiki/Algebra_lineare" title="Algebra lineare">algebra lineare</a>, un <b>autovettore</b> di una funzione tra <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazi vettoriali</a> è un <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettore</a> non <a href="/wiki/Vettore_nullo" title="Vettore nullo">nullo</a> la cui <a href="/wiki/Immagine_(matematica)" title="Immagine (matematica)">immagine</a> è il vettore stesso moltiplicato per uno scalare detto <b>autovalore</b>.<sup id="cite_ref-def_1-0" class="reference"><a href="#cite_note-def-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Se la funzione è <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">lineare</a>, gli autovettori aventi in comune lo stesso autovalore, insieme con il vettore nullo, formano uno <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</a>, detto <b>autospazio</b>.<sup id="cite_ref-autospazio_2-0" class="reference"><a href="#cite_note-autospazio-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> La nozione di autovettore viene generalizzata dal concetto di <a href="/w/index.php?title=Vettore_radicale&action=edit&redlink=1" class="new" title="Vettore radicale (la pagina non esiste)">vettore radicale</a> o <i>autovettore generalizzato</i>. </p><p>I concetti di autovettore e autovalore sono utilizzati in molti settori della matematica e della <a href="/wiki/Fisica" title="Fisica">fisica</a>; il problema della ricerca degli autovalori di una funzione lineare corrisponde alla sua <a href="/wiki/Diagonalizzabilit%C3%A0" title="Diagonalizzabilità">diagonalizzazione</a>. Se un autovettore è una funzione, si parla di <a href="/wiki/Autofunzione" title="Autofunzione">autofunzione</a>; per esempio in <a href="/wiki/Meccanica_classica" title="Meccanica classica">meccanica classica</a> è molto comune considerare la <a href="/wiki/Funzione_esponenziale" title="Funzione esponenziale">funzione esponenziale</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\lambda }(x)=e^{\lambda x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\lambda }(x)=e^{\lambda x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b61d7961af0b0bfd5fd2f8cad2d505775e49d6e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.781ex; height:3.176ex;" alt="{\displaystyle f_{\lambda }(x)=e^{\lambda x}}"></span> come autofunzione della <a href="/wiki/Derivata" title="Derivata">derivata</a>. Formalismi di questo tipo consentono di descrivere molti problemi relativi ad un sistema fisico: ad esempio, i <a href="/wiki/Oscillatore_armonico" class="mw-redirect" title="Oscillatore armonico">modi di vibrazione</a> di un <a href="/wiki/Corpo_rigido" title="Corpo rigido">corpo rigido</a> o i <a href="/wiki/Matrice_di_Fock" title="Matrice di Fock">livelli energetici</a> degli <a href="/wiki/Orbitale_atomico" title="Orbitale atomico">orbitali atomici</a> e <a href="/wiki/Orbitale_molecolare" title="Orbitale molecolare">molecolari</a> sono associati ad autovettori (<a href="/wiki/Autostato" title="Autostato">autostati</a>) di funzioni (<a href="/wiki/Osservabile" title="Osservabile">osservabili</a>) che ne determinano la dinamica. </p><p>Il termine autovettore è stato tradotto dalla parola <a href="/wiki/Lingua_tedesca" title="Lingua tedesca">tedesca</a> <i>Eigenvektor</i>, coniata da <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> nel <a href="/wiki/1904" title="1904">1904</a>. <i>Eigen</i> significa "proprio", "caratteristico". Analogamente il prefisso <i>auto-</i> usato nella versione italiana non è abbreviazione di "automatico", bensì è preso dal greco <i>autós</i> con significato <i>"di sé stesso</i>". Nella letteratura italiana si trova spesso l'autovettore indicato come <i>vettore proprio</i>, <i>vettore caratteristico</i> o <i>vettore latente</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduzione_informale">Introduzione informale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=1" title="Modifica la sezione Introduzione informale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=1" title="Edit section's source code: Introduzione informale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rotation_illustration2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Rotation_illustration2.svg/220px-Rotation_illustration2.svg.png" decoding="async" width="220" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Rotation_illustration2.svg/330px-Rotation_illustration2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Rotation_illustration2.svg/440px-Rotation_illustration2.svg.png 2x" data-file-width="1005" data-file-height="897" /></a><figcaption>Esempio di trasformazione lineare: rotazione di una figura piana intorno a un punto O.</figcaption></figure> <p>Il <a href="/wiki/Piano_cartesiano" class="mw-redirect" title="Piano cartesiano">piano cartesiano</a> e lo <a href="/wiki/Spazio_euclideo" title="Spazio euclideo">spazio euclideo</a> sono esempi particolari di <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazi vettoriali</a>: ogni punto dello spazio può essere descritto tramite un <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettore</a>, rappresentato graficamente da un segmento che collega l'origine al punto. In uno spazio vettoriale è possibile effettuare <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazioni lineari</a> sui vettori che lo costituiscono: esempi di trasformazioni lineari sono le <a href="/wiki/Rotazione_(matematica)" title="Rotazione (matematica)">rotazioni</a>, le <a href="/wiki/Omotetia" title="Omotetia">omotetie</a> (che consentono a un vettore di essere amplificato o contratto) e le <a href="/wiki/Riflessione_(geometria)" title="Riflessione (geometria)">riflessioni</a> (che consentono a un vettore di essere trasformato nel suo speculare rispetto a un punto, retta o piano assegnati). </p><p>Un autovettore per la trasformazione lineare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> è un vettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b9d8377af2a91280f8386a7d04d3cfc5cba4a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.672ex; height:2.676ex;" alt="{\displaystyle \mathbf {v} \neq 0}"></span> che a seguito dell'applicazione di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> non cambia la sua direzione, limitandosi ad essere moltiplicato per uno scalare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, il rispettivo autovalore. Il vettore può quindi soltanto cambiare modulo (venendo amplificato o contratto) e verso (venendo ribaltato): </p> <ul><li>se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea25afc0351140f919cf791c49c1964b8b081de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda >0}"></span> il verso di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> rimane inalterato, mentre se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8153323bd5bcdfe0c76df0db7ba3c4b3686ead55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda <0}"></span> il verso di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> cambia</li> <li>se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\lambda |=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\lambda |=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f99c7b1dc3696f6e900ea51d946e4da3915f7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.91ex; height:2.843ex;" alt="{\displaystyle |\lambda |=1}"></span> il modulo di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> rimane inalterato, se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\lambda |>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\lambda |>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/570bc8c89bbe3ab6a22dead01eb019d457280192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.91ex; height:2.843ex;" alt="{\displaystyle |\lambda |>1}"></span> il modulo cresce, se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\lambda |<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\lambda |<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e63f0da73af66e38ef253b0a16d7001fe1c314b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.91ex; height:2.843ex;" alt="{\displaystyle |\lambda |<1}"></span> decresce.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Standing_wave.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standing_wave.gif/260px-Standing_wave.gif" decoding="async" width="260" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standing_wave.gif/390px-Standing_wave.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standing_wave.gif/520px-Standing_wave.gif 2x" data-file-width="750" data-file-height="250" /></a><figcaption>Un'<a href="/wiki/Onda_(fisica)" class="mw-redirect" title="Onda (fisica)">onda</a> stazionaria in una corda fissata agli estremi è una <a href="/wiki/Autofunzione" title="Autofunzione">autofunzione</a> della trasformazione data dallo scorrere del tempo.</figcaption></figure> <p>Autovettori e autovalori sono definiti e usati in matematica e fisica nell'ambito di spazi vettoriali più complessi e astratti di quello tridimensionale della fisica classica. Questi spazi possono avere <a href="/wiki/Dimensione_di_Hamel" class="mw-redirect" title="Dimensione di Hamel">dimensione</a> maggiore di 3 o addirittura infinita (un esempio è dato dallo <a href="/wiki/Spazio_di_Hilbert" title="Spazio di Hilbert">spazio di Hilbert</a>). Anche le possibili posizioni di una <a href="/wiki/Corda_vibrante" title="Corda vibrante">corda vibrante</a> in una chitarra formano uno spazio di questo tipo: una vibrazione della corda è quindi interpretata come trasformazione di questo spazio e i suoi autovettori (più precisamente, le sue <a href="/wiki/Autofunzione" title="Autofunzione">autofunzioni</a>) sono le <a href="/wiki/Onda_stazionaria" title="Onda stazionaria">onde stazionarie</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definizione">Definizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=2" title="Modifica la sezione Definizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=2" title="Edit section's source code: Definizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> uno <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</a> su un <a href="/wiki/Campo_(matematica)" title="Campo (matematica)">campo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span>, che può essere ad esempio il campo dei <a href="/wiki/Numeri_reali" class="mw-redirect" title="Numeri reali">numeri reali</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> o il campo dei <a href="/wiki/Numeri_complessi" class="mw-redirect" title="Numeri complessi">complessi</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> un <a href="/wiki/Endomorfismo" title="Endomorfismo">endomorfismo</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, cioè una <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazione lineare</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\colon V\to V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\colon V\to V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf8f6fb35d4bbb18b579787cfec47818d48506a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.506ex; height:2.509ex;" alt="{\displaystyle T\colon V\to V,}"></span></dd></dl> <p>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> è un <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettore</a> non nullo in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> è uno <a href="/wiki/Grandezza_scalare" title="Grandezza scalare">scalare</a> tali che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c03cf705786a3d5f73ca46df287ae6739082160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}"></span></dd></dl> <p>allora <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> è un autovettore della trasformazione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> è il suo autovalore.<sup id="cite_ref-def_1-1" class="reference"><a href="#cite_note-def-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Poiché <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è lineare, se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> è un autovettore con autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, allora ogni multiplo non-nullo di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> è anch'esso un autovettore con lo stesso autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>. Infatti, detto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"></span> un qualsiasi vettore tale che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =k\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =k\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d7544bb5a2582a6faf5358d9865d3e919f4830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.206ex; height:2.176ex;" alt="{\displaystyle \mathbf {u} =k\mathbf {v} }"></span>, con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3846561ca971f4b17e10153d1f996e08e5ac192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.118ex; height:2.176ex;" alt="{\displaystyle k\in K}"></span>, si avrà <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {u} )=T(k\mathbf {v} )=kT(\mathbf {v} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {u} )=T(k\mathbf {v} )=kT(\mathbf {v} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd28f4f4d7ba19f142f67bc83bcf4bad1a5916f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.263ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {u} )=T(k\mathbf {v} )=kT(\mathbf {v} )}"></span> poiché <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è lineare. Ma essendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd39fa2822cd6c014b0636395aae1fad4e2b4cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.721ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} }"></span>, si ha che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {u} )=kT(\mathbf {v} )=k\lambda \mathbf {v} =\lambda k\mathbf {v} =\lambda \mathbf {u} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {u} )=kT(\mathbf {v} )=k\lambda \mathbf {v} =\lambda k\mathbf {v} =\lambda \mathbf {u} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7bc35d617c86e39b5e576594f0eed36087c185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.835ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {u} )=kT(\mathbf {v} )=k\lambda \mathbf {v} =\lambda k\mathbf {v} =\lambda \mathbf {u} ,}"></span></dd></dl> <p>cioè <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {u} )=\lambda \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {u} )=\lambda \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08f4d8c432980a92476ea5126e43e9a535572a05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.87ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {u} )=\lambda \mathbf {u} }"></span>. </p><p>Più in generale, gli autovettori aventi lo stesso fissato autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, insieme al vettore nullo, generano un <a href="/wiki/Sottospazio_vettoriale" title="Sottospazio vettoriale">sottospazio</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> chiamato l'autospazio relativo all'autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, solitamente indicato con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99e54af252e9a82c9d14480301e5783f79357b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.546ex; height:2.509ex;" alt="{\displaystyle V_{\lambda }}"></span>.<sup id="cite_ref-autospazio_2-1" class="reference"><a href="#cite_note-autospazio-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Lo <a href="/wiki/Spettro_(matematica)" title="Spettro (matematica)">spettro</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è l'insieme dei suoi autovalori. Il <a href="/wiki/Raggio_spettrale" title="Raggio spettrale">raggio spettrale</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è l'<a href="/wiki/Estremo_superiore_e_estremo_inferiore" title="Estremo superiore e estremo inferiore">estremo superiore</a> dei moduli dei suoi autovalori. </p> <div class="mw-heading mw-heading3"><h3 id="Descrizione_matriciale_e_autovettore_sinistro">Descrizione matriciale e autovettore sinistro</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=3" title="Modifica la sezione Descrizione matriciale e autovettore sinistro" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=3" title="Edit section's source code: Descrizione matriciale e autovettore sinistro"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nel caso in cui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> sia di dimensione <a href="/wiki/Insieme_finito" title="Insieme finito">finita</a>, per ogni scelta di <a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">basi</a> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è associata univocamente una <a href="/wiki/Matrice_(matematica)" class="mw-redirect" title="Matrice (matematica)">matrice</a>, detta <a href="/wiki/Matrice_di_trasformazione" title="Matrice di trasformazione">matrice di trasformazione</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Si può pertanto parlare di una funzione lineare sia in termini di funzione (trasformazione) che di matrice, ed il formalismo matriciale viene spesso utilizzato per la ricerca di autovettori e autovalori. </p><p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span> il vettore delle coordinate di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> rispetto a una base e sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> la matrice di trasformazione rappresentante <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> rispetto alla medesima base. Si ha che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span> è detto autovettore di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> se esiste uno scalare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> detto autovalore tale che:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} =\lambda \mathbf {x} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\lambda \mathbf {x} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769b8a79fcdd0d5380df6359c5d558159ac0dc3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.666ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\lambda \mathbf {x} .}"></span></dd></dl> <p>In particolare, gli autovalori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> non dipendono dalla base scelta. </p><p>Il vettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span> è detto <i>autovettore destro</i>, in quanto un vettore non nullo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00f4fe90eb178382625be6a05790d826e051f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.762ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{L}}"></span> è detto <i>autovettore sinistro</i> se esiste <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> tale che:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \mathbf {x} _{L}^{H}=\mathbf {x} _{L}^{H}A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msubsup> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msubsup> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \mathbf {x} _{L}^{H}=\mathbf {x} _{L}^{H}A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9212d876d849e93b78fe0abbf78f6c0cb653db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.049ex; height:3.176ex;" alt="{\displaystyle \lambda \mathbf {x} _{L}^{H}=\mathbf {x} _{L}^{H}A,}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{L}^{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{L}^{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b62e2ff779f1ce1b81db5ca4fa3e76131f874d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.102ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} _{L}^{H}}"></span> è il vettore <a href="/wiki/Matrice_trasposta" title="Matrice trasposta">trasposto</a> <a href="/wiki/Complesso_coniugato" title="Complesso coniugato">complesso coniugato</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00f4fe90eb178382625be6a05790d826e051f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.762ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{L}}"></span>. Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00f4fe90eb178382625be6a05790d826e051f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.762ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{L}}"></span> è autovettore sinistro di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> con autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>, allora <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00f4fe90eb178382625be6a05790d826e051f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.762ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{L}}"></span> è anche autovettore destro della <a href="/wiki/Matrice_trasposta_coniugata" title="Matrice trasposta coniugata">matrice trasposta coniugata</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8904d54cf6b5808177fdfbd13c2b73966d8e0ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.676ex;" alt="{\displaystyle A^{H}}"></span> con autovalore il complesso coniugato <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b40caf100572a9cc91ebf6ff7079ca8dd9eeb59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.509ex;" alt="{\displaystyle {\bar {\lambda }}}"></span>. </p><p>Senza ulteriori precisazioni, per "autovettore" si intende l'autovettore destro. </p> <div class="mw-heading mw-heading3"><h3 id="Autofunzioni">Autofunzioni</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=4" title="Modifica la sezione Autofunzioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=4" title="Edit section's source code: Autofunzioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Autofunzione" title="Autofunzione">Autofunzione</a></b>.</span></div> </div> <p>Spesso gli autovettori sono a loro volta funzioni, e in tal caso si parla di autofunzioni di un <a href="/wiki/Operatore_(matematica)" title="Operatore (matematica)">operatore</a>. Un esempio molto significativo in matematica e fisica è quello dell'autofunzione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{k}(x)=e^{kx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{k}(x)=e^{kx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced5c9a38c60f27e282c3f2f907d16125978f8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.578ex; height:3.176ex;" alt="{\displaystyle f_{k}(x)=e^{kx}}"></span></dd></dl> <p>dell'<a href="/wiki/Operatore_differenziale" title="Operatore differenziale">operatore differenziale</a> <a href="/wiki/Derivata" title="Derivata">derivata</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}={\frac {d}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}={\frac {d}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/161d37f9434a3700881130e3b2156d7f08921907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.384ex; height:5.509ex;" alt="{\displaystyle {\mathcal {A}}={\frac {d}{dx}}}"></span></dd></dl> <p>a cui corrisponde l'autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a28cfdf30808b2d95d2b89990ecbf39cfa8c468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.665ex; height:2.176ex;" alt="{\displaystyle \lambda =k}"></span> in quanto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}(e^{kx})=ke^{kx}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}(e^{kx})=ke^{kx}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ca303f4d687020cedcdf9e74da7485e86603dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.894ex; height:3.176ex;" alt="{\displaystyle {\mathcal {A}}(e^{kx})=ke^{kx}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Polinomio_caratteristico">Polinomio caratteristico</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=5" title="Modifica la sezione Polinomio caratteristico" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=5" title="Edit section's source code: Polinomio caratteristico"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Polinomio_caratteristico" title="Polinomio caratteristico">Polinomio caratteristico</a></b>.</span></div> </div> <p>Si definisce <a href="/wiki/Polinomio" title="Polinomio">polinomio</a> caratteristico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c61e08ebb244e9238530b5fe0c407b0fe4fe842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.423ex; height:2.843ex;" alt="{\displaystyle p(\lambda )}"></span> nella variabile <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> associato a una matrice quadrata <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> il <a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">determinante</a>:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\lambda )=\det(A-\lambda I),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\lambda )=\det(A-\lambda I),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/376b14386d2af5027edd5df1c699db1ffaea4086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:20.318ex; height:2.843ex;" alt="{\displaystyle p(\lambda )=\det(A-\lambda I),}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> è la <a href="/wiki/Matrice_identit%C3%A0" title="Matrice identità">matrice identità</a> con lo stesso numero di righe di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. Le radici del polinomio caratteristico sono tutti gli autovalori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Due matrici che rappresentano un <a href="/wiki/Endomorfismo" title="Endomorfismo">endomorfismo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> di uno spazio vettoriale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> a <a href="/wiki/Dimensione_(spazio_vettoriale)" title="Dimensione (spazio vettoriale)">dimensione</a> finita sono <a href="/wiki/Similitudine_fra_matrici" class="mw-redirect" title="Similitudine fra matrici">simili</a>, e in particolare hanno il medesimo polinomio caratteristico, e dunque gli stessi autovalori. Si tratta di uno strumento di grande importanza, che ha permesso di sviluppare un metodo generale per l'individuazione di autovalori e autovettori di un endomorfismo nel caso in cui lo spazio vettoriale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> abbia <a href="/wiki/Dimensione" title="Dimensione">dimensione</a> finita.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Il polinomio permette inoltre di stabilire l'esistenza di autovalori e autovettori per un'applicazione lineare: </p> <ul><li>Il polinomio caratteristico di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> ha grado <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, e quindi ha al più <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> radici: segue che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> ha al più <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> autovalori distinti.</li> <li>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span> è <a href="/wiki/Campo_algebricamente_chiuso" title="Campo algebricamente chiuso">algebricamente chiuso</a> allora il polinomio caratteristico ha sempre almeno una radice: segue che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> ha almeno un autovalore, e quindi anche almeno un autovettore.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Nel caso reale questo non succede sempre, ad esempio si possono trovare autovalori complessi.</li> <li>Se la dimensione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> è dispari e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6419d3aa99701ca996737b17a5e1174d53e6c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {R} }"></span> è il campo dei <a href="/wiki/Numeri_reali" class="mw-redirect" title="Numeri reali">numeri reali</a>, il polinomio caratteristico ha grado dispari, e quindi ha sempre almeno una radice reale. Ad esempio, ogni endomorfismo di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> ha almeno un autovettore.</li> <li>Inoltre se il polinomio caratteristico di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è completamente fattorizzabile allora <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è triangolabile, ossia esiste base di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> tale per cui la matrice associata è una <a href="/wiki/Matrice_triangolare" title="Matrice triangolare">matrice triangolare</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Diagonalizzabilità"><span id="Diagonalizzabilit.C3.A0"></span>Diagonalizzabilità</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=6" title="Modifica la sezione Diagonalizzabilità" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=6" title="Edit section's source code: Diagonalizzabilità"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Diagonalizzabilit%C3%A0" title="Diagonalizzabilità">Diagonalizzabilità</a></b>.</span></div> </div> <p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> un <a href="/wiki/Endomorfismo" title="Endomorfismo">endomorfismo</a> di uno <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, cioè una <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazione lineare</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:V\to V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fabb548ced097e03f45cf54dbb066d23d010f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.762ex; height:2.176ex;" alt="{\displaystyle T:V\to V}"></span>. Si dice che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è diagonalizzabile se esiste una <a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">base</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> rispetto alla quale la matrice che <a href="/wiki/Matrice_di_trasformazione" title="Matrice di trasformazione">rappresenta</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è <a href="/wiki/Matrice_diagonale" title="Matrice diagonale">diagonale</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In particolare, la base che diagonalizza <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è composta da suoi autovettori. </p><p>In modo equivalente, una <a href="/wiki/Matrice_quadrata" title="Matrice quadrata">matrice quadrata</a> è diagonalizzabile se è <a href="/wiki/Similitudine_fra_matrici" class="mw-redirect" title="Similitudine fra matrici">simile</a> a una <a href="/wiki/Matrice_diagonale" title="Matrice diagonale">matrice diagonale</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> La matrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è quindi diagonalizzabile nel campo di appartenenza se esiste una matrice invertibile <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> tale che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{-1}TP={\begin{pmatrix}\lambda _{1}\\&\lambda _{2}\\&&\ddots \\&&&\lambda _{n}\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>T</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>⋱</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd /> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{-1}TP={\begin{pmatrix}\lambda _{1}\\&\lambda _{2}\\&&\ddots \\&&&\lambda _{n}\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c863d2be4d6404ce1d8337a7511b83bda89820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:33.439ex; height:14.176ex;" alt="{\displaystyle P^{-1}TP={\begin{pmatrix}\lambda _{1}\\&\lambda _{2}\\&&\ddots \\&&&\lambda _{n}\end{pmatrix}},}"></span></dd></dl> <p>ossia: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TP=P{\begin{pmatrix}\lambda _{1}\\&\lambda _{2}\\&&\ddots \\&&&\lambda _{n}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>P</mi> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>⋱</mo> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd /> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TP=P{\begin{pmatrix}\lambda _{1}\\&\lambda _{2}\\&&\ddots \\&&&\lambda _{n}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b11e499d295fd9ac222ff53edf6bc95436689b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:31.03ex; height:14.176ex;" alt="{\displaystyle TP=P{\begin{pmatrix}\lambda _{1}\\&\lambda _{2}\\&&\ddots \\&&&\lambda _{n}\end{pmatrix}}.}"></span></dd></dl> <p>Scrivendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> in termini dei vettori colonna: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\begin{pmatrix}\mathbf {P} ^{1}&\mathbf {P} ^{2}&\cdots &\mathbf {P} ^{n}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\begin{pmatrix}\mathbf {P} ^{1}&\mathbf {P} ^{2}&\cdots &\mathbf {P} ^{n}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edabd8fd2c03e943ee836cc163f2929c4eb22560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.55ex; height:3.176ex;" alt="{\displaystyle P={\begin{pmatrix}\mathbf {P} ^{1}&\mathbf {P} ^{2}&\cdots &\mathbf {P} ^{n}\end{pmatrix}}.}"></span></dd></dl> <p>La precedente relazione diventa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\mathbf {P} ^{i}=\lambda _{i}\mathbf {P} ^{i}\qquad (i=1,2,\cdots ,n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\mathbf {P} ^{i}=\lambda _{i}\mathbf {P} ^{i}\qquad (i=1,2,\cdots ,n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d52df2880ae4da339eb88b0e63ab70178e06400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.076ex; height:3.176ex;" alt="{\displaystyle T\mathbf {P} ^{i}=\lambda _{i}\mathbf {P} ^{i}\qquad (i=1,2,\cdots ,n).}"></span></dd></dl> <p>I vettori colonna di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> sono dunque autovettori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, e i corrispondenti elementi della matrice diagonale sono i rispettivi autovalori. L'invertibilità di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> implica inoltre l'indipendenza lineare degli autovettori, che formano una base dello spazio. </p> <div class="mw-heading mw-heading3"><h3 id="Il_teorema_spettrale">Il teorema spettrale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=7" title="Modifica la sezione Il teorema spettrale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=7" title="Edit section's source code: Il teorema spettrale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Teorema_spettrale" title="Teorema spettrale">Teorema spettrale</a></b>.</span></div> </div> <p>Nel caso complesso finito-dimensionale il <a href="/wiki/Teorema_spettrale" title="Teorema spettrale">teorema spettrale</a> afferma che l'endomorfismo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è <a href="/wiki/Operatore_normale" title="Operatore normale">normale</a> <a href="/wiki/Se_e_solo_se" title="Se e solo se">se e solo se</a> esiste una <a href="/wiki/Base_ortonormale" title="Base ortonormale">base ortonormale</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> fatta di suoi <a href="/wiki/Autovettore" class="mw-redirect" title="Autovettore">autovettori</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> In tal caso la matrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> è <a href="/wiki/Matrice_unitaria" title="Matrice unitaria">unitaria</a>. Questo fondamentale risultato fornisce le condizioni per cui sia possibile diagonalizzare un <a href="/wiki/Operatore_lineare_continuo" title="Operatore lineare continuo">operatore lineare</a> rispetto a una base ortonormale: nel caso finito-dimensionale, quando questo risulta possibile succede che ad autovalori distinti corrispondono autovettori mutuamente ortogonali, e pertanto gli <a href="/wiki/Autospazio" class="mw-redirect" title="Autospazio">autospazi</a> sono in <a href="/wiki/Somma_diretta" title="Somma diretta">somma diretta</a>. </p><p>La decomposizione spettrale è un caso particolare della <a href="/wiki/Decomposizione_di_Schur" title="Decomposizione di Schur">decomposizione di Schur</a>. È anche un caso particolare della <a href="/wiki/Decomposizione_ai_valori_singolari" title="Decomposizione ai valori singolari">decomposizione ai valori singolari</a>. Un operatore normale può, di conseguenza, essere scritto come una combinazione lineare di proiettori ortogonali sugli autospazi, i cui coefficienti sono gli autovalori relativi a ogni autospazio. </p><p>Nel caso infinito-dimensionale la normalità, e in particolare l'<a href="/wiki/Operatore_autoaggiunto" title="Operatore autoaggiunto">autoaggiuntezza</a>, non garantisce la diagonalizzabilità. In generale un operatore normale non può essere più scritto come combinazione lineare di proiettori ortogonali. Tuttavia, attraverso una <a href="/wiki/Misura_a_valori_di_proiettore" title="Misura a valori di proiettore">misura a valori di proiettore</a> è possibile ottenere una scrittura integrale che permette di descrivere l'operatore in termini del suo <a href="/wiki/Spettro_(matematica)" title="Spettro (matematica)">spettro</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Spettro_di_un_operatore">Spettro di un operatore</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=8" title="Modifica la sezione Spettro di un operatore" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=8" title="Edit section's source code: Spettro di un operatore"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Spettro_(matematica)" title="Spettro (matematica)">Spettro (matematica)</a></b>.</span></div> </div> <p>In uno spazio di dimensione infinita la definizione di autovalore è identica al caso di dimensione finita. Tuttavia, il <a href="/wiki/Polinomio_caratteristico" title="Polinomio caratteristico">polinomio caratteristico</a> non è uno strumento disponibile in questo caso in quanto si rende necessario considerare ulteriori elementi dello spettro. </p><p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> un <a href="/wiki/Operatore_lineare_continuo" title="Operatore lineare continuo">operatore lineare</a> <a href="/wiki/Operatore_limitato" title="Operatore limitato">limitato</a> definito su uno <a href="/wiki/Spazio_di_Banach" title="Spazio di Banach">spazio di Banach</a> complesso <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. Si definisce <i>insieme risolvente</i> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> l'insieme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8068365c7aae72114e4fc7e997875be501023866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.648ex; height:2.843ex;" alt="{\displaystyle \rho (T)}"></span> dei <a href="/wiki/Numero_complesso" title="Numero complesso">numeri complessi</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> tali per cui l'operatore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda I-T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mi>I</mi> <mo>−</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda I-T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b2086dea5e2a80ceec37321f7e778abbfb99ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.004ex; height:2.343ex;" alt="{\displaystyle \lambda I-T}"></span> è invertibile, ovvero ha un <a href="/wiki/Funzione_inversa" title="Funzione inversa">inverso</a> che è un operatore lineare limitato. Si definisce <i>risolvente</i> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> la funzione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\lambda }(T)=(\lambda I-T)^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>I</mi> <mo>−</mo> <mi>T</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\lambda }(T)=(\lambda I-T)^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e65b5a0ab20d5d50897d5daaa5bc2b2d97362b09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.291ex; height:3.176ex;" alt="{\displaystyle R_{\lambda }(T)=(\lambda I-T)^{-1}.}"></span></dd></dl> <p>Lo spettro di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è l'insieme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9888a5496892a8c482ad465f48b839102187bd1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle \sigma (T)}"></span> dei numeri complessi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> che non appartengono all'insieme risolvente, ovvero tali per cui l'operatore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda I-T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mi>I</mi> <mo>−</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda I-T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b2086dea5e2a80ceec37321f7e778abbfb99ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.004ex; height:2.343ex;" alt="{\displaystyle \lambda I-T}"></span> non è invertibile.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Dal momento che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda I-T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mi>I</mi> <mo>−</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda I-T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b2086dea5e2a80ceec37321f7e778abbfb99ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.004ex; height:2.343ex;" alt="{\displaystyle \lambda I-T}"></span> è un <a href="/wiki/Operatore_lineare_continuo" title="Operatore lineare continuo">operatore lineare</a>, se il suo inverso esiste esso è lineare. Inoltre, per il <a href="/wiki/Teorema_del_grafico_chiuso" title="Teorema del grafico chiuso">teorema del grafico chiuso</a> l'inverso di un <a href="/wiki/Operatore_limitato" title="Operatore limitato">operatore lineare limitato</a> è limitato. Segue che l'insieme risolvente è l'insieme dei valori che rendono <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda I-T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mi>I</mi> <mo>−</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda I-T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b2086dea5e2a80ceec37321f7e778abbfb99ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.004ex; height:2.343ex;" alt="{\displaystyle \lambda I-T}"></span> bigettivo. </p><p>Lo spettro di un operatore non può essere vuoto, e si possono distinguere tre suoi sottoinsiemi disgiunti: </p> <ul><li>Si definisce <i>spettro puntuale</i> o <i>discreto</i> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> l'insieme degli autovalori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, ovvero i numeri complessi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> tali che:</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)=\lambda x,\qquad x\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>x</mi> <mo>,</mo> <mspace width="2em" /> <mi>x</mi> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)=\lambda x,\qquad x\neq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a5e0a4e82333c87e68e7871f759f171170363a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.475ex; height:2.843ex;" alt="{\displaystyle T(x)=\lambda x,\qquad x\neq 0.}"></span></dd></dl> <dl><dd>Gli autovalori sono quindi i numeri tali per cui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)-\lambda x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>λ</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)-\lambda x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a4170aa0a9d5e7cbd6f19413b1211fa241ca03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.561ex; height:2.843ex;" alt="{\displaystyle T(x)-\lambda x=0}"></span>, ossia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T-\lambda I)(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T-\lambda I)(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2f0ed7c2be57ac27edbea16cb9e69e0039db59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.213ex; height:2.843ex;" alt="{\displaystyle (T-\lambda I)(x)=0}"></span>: la funzione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T-\lambda I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T-\lambda I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1851957e0e104ea3a266ef2ff75e0530485755c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.004ex; height:2.343ex;" alt="{\displaystyle T-\lambda I}"></span> non è invertibile se il suo <a href="/wiki/Nucleo_(matematica)" title="Nucleo (matematica)">nucleo</a> non è costituito dal solo vettore nullo, ovvero esistono dei vettori <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> tali per cui esiste un <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> tale che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)-\lambda x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>λ</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)-\lambda x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a4170aa0a9d5e7cbd6f19413b1211fa241ca03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.561ex; height:2.843ex;" alt="{\displaystyle T(x)-\lambda x=0}"></span>. In modo equivalente, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> è autovalore di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> se e solo se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T-\lambda I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T-\lambda I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1851957e0e104ea3a266ef2ff75e0530485755c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.004ex; height:2.343ex;" alt="{\displaystyle T-\lambda I}"></span> non è iniettivo, oppure se e solo se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(T-\lambda I)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(T-\lambda I)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e5623c33dd8d7e880320dbc25989bba9529d8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.303ex; height:2.843ex;" alt="{\displaystyle \det(T-\lambda I)=0}"></span>.</dd></dl> <ul><li>Si definisce <i>spettro continuo</i> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> l'insieme dei numeri <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> tali per cui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda I-T)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>I</mi> <mo>−</mo> <mi>T</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda I-T)^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59b76101b55c8630978fb93f7487ce1b168e0cfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.146ex; height:3.176ex;" alt="{\displaystyle (\lambda I-T)^{-1}}"></span> non è limitato, pur essendo densamente definito.</li> <li>Si definisce <i>spettro residuo</i> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> l'insieme dei numeri <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> che non sono autovalori e tali per cui l'operatore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda I-T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mi>I</mi> <mo>−</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda I-T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b2086dea5e2a80ceec37321f7e778abbfb99ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.004ex; height:2.343ex;" alt="{\displaystyle \lambda I-T}"></span> non ha <a href="/wiki/Immagine_(matematica)" title="Immagine (matematica)">immagine</a> <a href="/wiki/Insieme_denso" title="Insieme denso">densa</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Operatori_aggiunti_e_autoaggiunti">Operatori aggiunti e autoaggiunti</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=9" title="Modifica la sezione Operatori aggiunti e autoaggiunti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=9" title="Edit section's source code: Operatori aggiunti e autoaggiunti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Operatore_aggiunto" title="Operatore aggiunto">Operatore aggiunto</a></b> e <b><a href="/wiki/Operatore_autoaggiunto" title="Operatore autoaggiunto">Operatore autoaggiunto</a></b>.</span></div> </div> <p>La definizione di <a href="/wiki/Operatore_aggiunto" title="Operatore aggiunto">operatore aggiunto</a> si diversifica a seconda che ci si trovi in uno <a href="/wiki/Spazio_di_Hilbert" title="Spazio di Hilbert">spazio di Hilbert</a> o in uno <a href="/wiki/Spazio_di_Banach" title="Spazio di Banach">spazio di Banach</a>. A causa di ciò, lo spettro di un operatore definito su uno spazio di Banach coincide con quello del suo aggiunto, mentre in uno spazio di Hilbert, denotando l'aggiunto di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5426ed7abdea7d2ff995ad0e01bc4ca62a273855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.343ex;" alt="{\displaystyle T^{*}}"></span>, si ha che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (T^{*})=\{\lambda :{\bar {\lambda }}\in \sigma (T)\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>λ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>∈</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (T^{*})=\{\lambda :{\bar {\lambda }}\in \sigma (T)\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30a1cf30d2b34a4524ebb3ac9b8f2731035a1f2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.247ex; height:3.009ex;" alt="{\displaystyle \sigma (T^{*})=\{\lambda :{\bar {\lambda }}\in \sigma (T)\}.}"></span></dd></dl> <p>Inoltre, se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> appartiene allo spettro residuo di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, allora <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> appartiene allo spettro puntuale dell'aggiunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b32d735f4fbb6eff3b35ed3dc1005a069d0b2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle T'}"></span>. Se invece <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> appartiene allo spettro puntuale di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, allora esso appartiene sia allo spettro puntuale e sia allo spettro residuo di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b32d735f4fbb6eff3b35ed3dc1005a069d0b2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle T'}"></span>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è autoaggiunto su uno spazio di Hilbert, si ha inoltre: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> non ha spettro residuo.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9888a5496892a8c482ad465f48b839102187bd1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle \sigma (T)}"></span> è un sottoinsieme di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>, ossia gli autovalori sono reali.</li> <li>Autovettori relativi ad autovalori distinti sono ortogonali.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Applicazioni">Applicazioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=10" title="Modifica la sezione Applicazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=10" title="Edit section's source code: Applicazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lo studio degli autovalori e autovettori relativi a una trasformazione lineare, che consiste nell'<i>autoteoria</i>, è una delle problematiche principali affrontate dall'<a href="/wiki/Algebra_lineare" title="Algebra lineare">algebra lineare</a>, e ha vastissime applicazioni in diversi ambiti della scienza. </p> <div class="mw-heading mw-heading3"><h3 id="Operatori_in_meccanica_quantistica">Operatori in meccanica quantistica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=11" title="Modifica la sezione Operatori in meccanica quantistica" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=11" title="Edit section's source code: Operatori in meccanica quantistica"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Postulati_della_meccanica_quantistica" title="Postulati della meccanica quantistica">Postulati della meccanica quantistica</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:HAtomOrbitals.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/HAtomOrbitals.png/220px-HAtomOrbitals.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/cf/HAtomOrbitals.png 1.5x" data-file-width="316" data-file-height="316" /></a><figcaption>Le <a href="/wiki/Funzione_d%27onda" title="Funzione d'onda">funzioni d'onda</a> associate agli stati di un <a href="/wiki/Elettrone" title="Elettrone">elettrone</a> in un <a href="/wiki/Atomo" title="Atomo">atomo</a> d'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a> sono gli autovettori sia della <a href="/wiki/Operatore_hamiltoniano" title="Operatore hamiltoniano">Hamiltoniana</a> dell'atomo di idrogeno sia del <a href="/wiki/Momento_angolare" title="Momento angolare">momento angolare</a>. Gli autovalori associati sono interpretati come le loro <a href="/wiki/Energia" title="Energia">energie</a> (crescenti dall'alto in basso <i>n</i>=1,2,3,...) e <a href="/wiki/Momento_angolare" title="Momento angolare">momenti angolari</a> (crescenti da sinistra a destra: <i>s</i>, <i>p</i>, <i>d</i>,...). Sono disegnati qui i quadrati dei valori assoluti delle autofunzioni. Aree più luminose corrispondono a densità di <a href="/wiki/Probabilit%C3%A0" title="Probabilità">probabilità</a> maggiori per la posizione in una misurazione. Il centro di ogni figura è il <a href="/wiki/Nucleo_atomico" title="Nucleo atomico">nucleo dell'atomo</a>, un <a href="/wiki/Protone" title="Protone">protone</a>.</figcaption></figure> <p>In <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">meccanica quantistica</a> ad un vettore (detto in tale contesto <a href="/wiki/Notazione_bra-ket" title="Notazione bra-ket">autoket</a>) si associa uno "stato" o <i>autostato</i> dell'oggetto considerato. In termini informali, per evidenziare il fatto che in generale non si conosce questo stato, lo si descrive come una <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> (o <a href="/wiki/Principio_di_sovrapposizione_(meccanica_quantistica)" title="Principio di sovrapposizione (meccanica quantistica)">sovrapposizione</a>) di autovettori (autostati) noti di un qualche operatore. L'operatore in questione "rappresenta" una certa <a href="/wiki/Osservabile" title="Osservabile">osservabile</a>, alla quale corrisponde una <a href="/wiki/Grandezza_fisica" title="Grandezza fisica">grandezza fisica</a>: ad esempio l'<a href="/wiki/Operatore_hamiltoniano" title="Operatore hamiltoniano">operatore hamiltoniano</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> è associato all'<a href="/wiki/Energia" title="Energia">energia</a> dell'oggetto. Se si sceglie di scrivere lo stato dell'oggetto tramite una combinazione lineare di autovettori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, a ogni autovettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi _{E}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi _{E}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5465beabfb2e825cddf66a43fc3f371d5bc2ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.552ex; height:2.843ex;" alt="{\displaystyle \left|\psi _{E}\right\rangle }"></span> è associato un possibile valore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> dell'energia dell'oggetto, che è il relativo autovalore: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left|\psi _{E}\right\rangle =E\left|\psi _{E}\right\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>|</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>=</mo> <mi>E</mi> <mrow> <mo>|</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left|\psi _{E}\right\rangle =E\left|\psi _{E}\right\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/875a3c812ab81804ebf773c1a2849714a1b252ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.851ex; height:2.843ex;" alt="{\displaystyle H\left|\psi _{E}\right\rangle =E\left|\psi _{E}\right\rangle .}"></span></dd></dl> <p>Una tale rappresentazione matriciale dell'<a href="/wiki/Equazione_di_Schr%C3%B6dinger" title="Equazione di Schrödinger">equazione di Schrödinger</a> indipendente dal tempo è possibile se, come spesso accade (ad esempio studiando gli stati legati), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d591fcd4d776cad11073ce439ba393b0fa15b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.001ex; height:2.509ex;" alt="{\displaystyle \psi _{E}}"></span> è una <a href="/wiki/Funzione_quadrato_sommabile" class="mw-redirect" title="Funzione quadrato sommabile">funzione quadrato sommabile</a>: tali funzioni formano uno <a href="/wiki/Spazio_di_Hilbert" title="Spazio di Hilbert">spazio di Hilbert</a> infinito-dimensionale con <a href="/wiki/Forma_sesquilineare" title="Forma sesquilineare">prodotto interno</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle |\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle |\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edea76d6c1e663040324895daca9c8b26f4d6b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle \left\langle |\right\rangle }"></span>. </p><p>L'operazione che tramite l'applicazione di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> restituisce uno degli autovalori è detta <i>misura</i>, e fa "collassare" o "precipitare" lo stato dell'oggetto in un autostato dell'osservabile che si sta misurando. La misura altera irrimediabilmente lo stato del sistema, che viene a trovarsi in un autostato ben preciso. L'insieme dei valori (autovalori) possibili per la misura di una grandezza osservabile è lo <a href="/wiki/Spettro_(matematica)" title="Spettro (matematica)">spettro</a> dell'operatore ad essa associato. Dovendo quantificare una grandezza fisica, è inoltre necessario che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> sia un <a href="/wiki/Operatore_autoaggiunto" title="Operatore autoaggiunto">operatore hermitiano</a>: in questo modo gli autovalori sono tutti <a href="/wiki/Numero_reale" title="Numero reale">reali</a>, e i suoi autostati (normalizzati) formano una <a href="/wiki/Base_ortonormale" title="Base ortonormale">base ortonormale</a> dello spazio. Grazie al prodotto interno <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle |\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle |\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edea76d6c1e663040324895daca9c8b26f4d6b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle \left\langle |\right\rangle }"></span> è possibile <a href="/wiki/Proiezione_(geometria)" title="Proiezione (geometria)">proiettare</a> l'autostato <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi _{E}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi _{E}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da5465beabfb2e825cddf66a43fc3f371d5bc2ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.552ex; height:2.843ex;" alt="{\displaystyle \left|\psi _{E}\right\rangle }"></span> sulla una base di autostati di un altro operatore, come la base di autovettori <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle x\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mi>x</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle x\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8cf0fb6c58ac9f466f7e67d6f228a3f7f075e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.881ex; height:2.843ex;" alt="{\displaystyle \left\langle x\right|}"></span> dell'<a href="/wiki/Operatore_posizione" title="Operatore posizione">operatore posizione</a>. La proiezione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{E}(x)=\left\langle x|\psi _{E}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{E}(x)=\left\langle x|\psi _{E}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686da4bc267adf62bf6891433a2fc366578440ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.32ex; height:2.843ex;" alt="{\displaystyle \Psi _{E}(x)=\left\langle x|\psi _{E}\right\rangle }"></span></dd></dl> <p>definisce la <a href="/wiki/Funzione_d%27onda" title="Funzione d'onda">funzione d'onda</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/344a64e415b94687c5551e2ce5b6f8464b93dd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.296ex; height:2.509ex;" alt="{\displaystyle \Psi _{E}}"></span>, una descrizione probabilistica della posizione dell'oggetto. La funzione d'onda <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{E}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{E}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf5c0fe36dd96ac8a79903dc30e8fcdf8431458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.435ex; height:2.843ex;" alt="{\displaystyle \Psi _{E}(x)}"></span> è dunque un'<a href="/wiki/Autofunzione" title="Autofunzione">autofunzione</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> corrispondente all'autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left\langle x|\psi _{E}\right\rangle =E\left\langle x|\psi _{E}\right\rangle ,\qquad H\Psi _{E}(x)=E\Psi _{E}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>⟨</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mi>E</mi> <mrow> <mo>⟨</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>H</mi> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left\langle x|\psi _{E}\right\rangle =E\left\langle x|\psi _{E}\right\rangle ,\qquad H\Psi _{E}(x)=E\Psi _{E}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9449fe22d1fc9516ff7484665e6aec6e1662b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.806ex; height:2.843ex;" alt="{\displaystyle H\left\langle x|\psi _{E}\right\rangle =E\left\langle x|\psi _{E}\right\rangle ,\qquad H\Psi _{E}(x)=E\Psi _{E}(x).}"></span></dd></dl> <p>Il prodotto interno nello spazio di Hilbert è inoltre dato da: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle \psi _{1}|\psi _{2}\right\rangle =\int _{D}\left\langle \psi _{1}|x\right\rangle \left\langle x|\psi _{2}\right\rangle {\mbox{d}}x=\int _{D}\Psi _{1}^{*}(x)\Psi _{2}(x){\mbox{d}}x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> </mrow> <mo>⟩</mo> </mrow> <mrow> <mo>⟨</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>d</mtext> </mstyle> </mrow> <mi>x</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>d</mtext> </mstyle> </mrow> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \psi _{1}|\psi _{2}\right\rangle =\int _{D}\left\langle \psi _{1}|x\right\rangle \left\langle x|\psi _{2}\right\rangle {\mbox{d}}x=\int _{D}\Psi _{1}^{*}(x)\Psi _{2}(x){\mbox{d}}x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99f8aa2428916e0e555732c50f9249a3eb012c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.708ex; height:5.676ex;" alt="{\displaystyle \left\langle \psi _{1}|\psi _{2}\right\rangle =\int _{D}\left\langle \psi _{1}|x\right\rangle \left\langle x|\psi _{2}\right\rangle {\mbox{d}}x=\int _{D}\Psi _{1}^{*}(x)\Psi _{2}(x){\mbox{d}}x,}"></span></dd></dl> <p>dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></span> indica la <a href="/wiki/Numero_complesso" title="Numero complesso">coniugazione complessa</a>. Questo limita la possibilità di scelta dello spazio di Hilbert allo spazio delle funzioni a quadrato integrabile sul dominio scelto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, che può al limite essere tutto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Teoria_dei_numeri">Teoria dei numeri</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=12" title="Modifica la sezione Teoria dei numeri" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=12" title="Edit section's source code: Teoria dei numeri"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lo studio degli autovalori di una matrice ha importanti applicazioni anche nella <a href="/wiki/Teoria_dei_numeri" title="Teoria dei numeri">teoria dei numeri</a>. In particolare, si congettura che alcune statistiche sugli zeri non banali della <a href="/wiki/Funzione_zeta_di_Riemann" title="Funzione zeta di Riemann">funzione zeta di Riemann</a>, quali ad esempio quelle sulla distanza tra zeri consecutivi, siano le stesse di quelle relative alle <a href="/wiki/Matrice_hermitiana" title="Matrice hermitiana">matrici hermitiane</a> aleatorie (rispetto alla <a href="/wiki/Misura_di_Haar" title="Misura di Haar">misura di Haar</a>) di dimensione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> al <a href="/wiki/Limite_(matematica)" title="Limite (matematica)">tendere</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> all'infinito. Inoltre, è stato congetturato che anche la distribuzione dei valori della funzione zeta di Riemann sia ben approssimata, in media, dai valori assunti dal <a href="/wiki/Polinomio_caratteristico" title="Polinomio caratteristico">polinomio caratteristico</a> di tali matrici. Analoghe considerazioni si possono fare su altre famiglie di <a href="/wiki/Funzione_speciale" title="Funzione speciale">funzioni speciali</a>, quali ad esempio le <a href="/wiki/Funzioni_L_di_Dirichlet" class="mw-redirect" title="Funzioni L di Dirichlet">funzioni L di Dirichlet</a>, coinvolgendo anche altre famiglie di matrici aleatorie, come ad esempio le matrici <a href="/wiki/Matrice_simplettica" title="Matrice simplettica">simplettiche</a> o <a href="/wiki/Matrice_ortogonale" title="Matrice ortogonale">ortogonali</a>. Tale connessione ha avuto come risultato un fiorire di una serie di nuove congetture in teoria dei numeri.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Autofacce">Autofacce</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=13" title="Modifica la sezione Autofacce" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=13" title="Edit section's source code: Autofacce"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Eigenfaces.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Eigenfaces.png/220px-Eigenfaces.png" decoding="async" width="220" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Eigenfaces.png/330px-Eigenfaces.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/67/Eigenfaces.png 2x" data-file-width="357" data-file-height="426" /></a><figcaption>Le autofacce sono esempi di autovettori.</figcaption></figure> <p>Nella <a href="/wiki/Elaborazione_digitale_delle_immagini" title="Elaborazione digitale delle immagini">elaborazione digitale delle immagini</a>, le immagini di facce possono essere viste come vettori le cui componenti sono la luminosità dei singoli <a href="/wiki/Pixel" title="Pixel">pixel</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> La dimensione dello spazio vettoriale in cui sono ambientati è pari al numero di pixel, e gli autovettori di una particolare matrice, detta <a href="/wiki/Matrice_delle_covarianze" title="Matrice delle covarianze">matrice di covarianza</a>, sono chiamati <i>autofacce</i>. Essi sono molto utili per esprimere ogni faccia come una <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> di queste autofacce, e sono quindi anche un ottimo strumento di <a href="/wiki/Compressione_dei_dati" title="Compressione dei dati">compressione dei dati</a> per memorizzare e identificare un alto numero di facce. </p> <div class="mw-heading mw-heading3"><h3 id="Tensore_d'inerzia"><span id="Tensore_d.27inerzia"></span>Tensore d'inerzia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=14" title="Modifica la sezione Tensore d'inerzia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=14" title="Edit section's source code: Tensore d'inerzia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Meccanica_(fisica)" title="Meccanica (fisica)">meccanica</a>, gli autovettori del <a href="/wiki/Momento_di_inerzia" title="Momento di inerzia">tensore di inerzia</a> definiscono gli assi principali di un corpo rigido. Il tensore di inerzia è una quantità chiave, necessaria per determinare la rotazione di un corpo rigido intorno al suo <a href="/wiki/Centro_di_massa" title="Centro di massa">baricentro</a>. Gli autovettori del tensore delle deformazioni definiscono gli assi principali di deformazione. </p> <div class="mw-heading mw-heading2"><h2 id="Esempi">Esempi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=15" title="Modifica la sezione Esempi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=15" title="Edit section's source code: Esempi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Esempi_nel_piano">Esempi nel piano</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=16" title="Modifica la sezione Esempi nel piano" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=16" title="Edit section's source code: Esempi nel piano"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fra le trasformazioni del <a href="/wiki/Piano_cartesiano" class="mw-redirect" title="Piano cartesiano">piano cartesiano</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> si possono distinguere i seguenti casi speciali: </p> <ul><li><a href="/wiki/Rotazione_(matematica)" title="Rotazione (matematica)">Rotazione</a> antioraria di angolo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>. Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> non è un multiplo intero di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> non esiste alcun autovettore, infatti ogni vettore viene ruotato e cambia di direzione. Se invece <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =k\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>k</mi> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =k\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9290745d3bf782c8057b06846dc7137b968ecc20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.732ex; height:2.176ex;" alt="{\displaystyle \theta =k\pi }"></span>, con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> intero dispari, ogni vettore viene trasformato nel suo opposto, quindi ogni vettore non nullo è autovettore della rotazione con autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span>. Se invece <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> è pari la trasformazione non è altro che l'identità, per cui ogni vettore non nullo è autovettore con autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"></span>.</li></ul> <dl><dd>La rotazione può essere <a href="/wiki/Matrice_di_trasformazione" title="Matrice di trasformazione">rappresentata</a> dalla seguente matrice:</dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/167b86e9229a8d87c0e76f921fb8d03b1edb6966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.293ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}.}"></span></dd></dl></dd></dl> <ul><li><a href="/wiki/Riflessione_(geometria)" title="Riflessione (geometria)">Riflessione</a> rispetto a una retta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> passante per l'origine. I vettori in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> restano fermi e sono quindi autovettori con autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>, mentre quelli della retta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> perpendicolare a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> e passante per l'origine vengono ribaltati, e quindi sono autovettori con autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"></span>. Non esistono altri autovettori.</li></ul> <dl><dd>La riflessione, nel caso di retta r orizzontale, può essere <a href="/wiki/Matrice_di_trasformazione" title="Matrice di trasformazione">rappresentata</a> dalla seguente matrice:</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0\\0&-1\\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0\\0&-1\\\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09c39dfd654205217eae3bca33083639a9010d7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.309ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0\\0&-1\\\end{bmatrix}}.}"></span></dd></dl> <ul><li><a href="/wiki/Omotetia" title="Omotetia">Omotetia</a>. Ogni vettore viene moltiplicato per uno scalare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> e quindi tutti i vettori non nulli sono autovettori con autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span>.</li></ul> <dl><dd>L'omotetia può essere <a href="/wiki/Matrice_di_trasformazione" title="Matrice di trasformazione">rappresentata</a> dalla seguente matrice:</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\lambda &0\\0&\lambda \\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>λ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>λ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\lambda &0\\0&\lambda \\\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/805d4fecc40713360fae373393cf0c38faa9dd2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.886ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\lambda &0\\0&\lambda \\\end{bmatrix}}.}"></span></dd></dl> <ul><li><a href="/wiki/Proiezione_ortogonale" class="mw-redirect" title="Proiezione ortogonale">Proiezione ortogonale</a> su una retta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> passante per l'origine. I vettori su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> restano fermi e quindi sono autovettori con autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>, mentre i vettori sulla retta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> ortogonale a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> e passante per l'origine sono mappati tutti sull'origine e quindi sono autovettori con autovalore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>. Non ci sono altri autovettori.</li></ul> <dl><dd>La proiezione ortogonale può essere <a href="/wiki/Matrice_di_trasformazione" title="Matrice di trasformazione">rappresentata</a> dalla seguente matrice:</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0\\0&0\\\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>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 {\begin{bmatrix}1&0\\0&0\\\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d60c7562b84a8e6d8627e94beb3607a39c3c4bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.501ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&0\\0&0\\\end{bmatrix}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Esempi_nello_spazio">Esempi nello spazio</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=17" title="Modifica la sezione Esempi nello spazio" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=17" title="Edit section's source code: Esempi nello spazio"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Non tutte le trasformazioni del piano e dello spazio ricadono in una delle 4 tipologie viste negli esempi del piano sopra riportate. </p><p>In generale, un <a href="/wiki/Endomorfismo" title="Endomorfismo">endomorfismo</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> (cioè una <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazione lineare</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> in sé) è rappresentabile tramite una <a href="/wiki/Matrice_quadrata" title="Matrice quadrata">matrice quadrata</a> con <i>n</i> righe. Si consideri per esempio l'endomorfismo di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> indotto dalla matrice: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}\;0&1&-1\\\;1&1&\;0\\-1&0&\;1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}\;0&1&-1\\\;1&1&\;0\\-1&0&\;1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ffcb9c409cc6087c990b841d61f0e9c349c08bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.089ex; height:9.176ex;" alt="{\displaystyle A={\begin{bmatrix}\;0&1&-1\\\;1&1&\;0\\-1&0&\;1\end{bmatrix}}.}"></span></dd></dl> <p>Se si considera il vettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d33f5d498d528bd8c10edc8ac8c34347f32b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{1}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}={\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}={\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7047f4b9956730a5e070607e375f67cceda9385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:12.103ex; height:9.176ex;" alt="{\displaystyle v_{1}={\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}}"></span></dd></dl> <p>e si esegue la <a href="/wiki/Prodotto_fra_matrici" class="mw-redirect" title="Prodotto fra matrici">moltiplicazione fra matrice e vettore</a>, si vede che: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}={\begin{bmatrix}\;2\\\;2\\-2\end{bmatrix}}=2{\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}={\begin{bmatrix}\;2\\\;2\\-2\end{bmatrix}}=2{\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceff18042336255e2ac63bbfa643d314e4d75bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:30.217ex; height:9.176ex;" alt="{\displaystyle A{\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}={\begin{bmatrix}\;2\\\;2\\-2\end{bmatrix}}=2{\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}.}"></span></dd></dl> <p>Quindi l'endomorfismo rappresentato da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> ha un autovettore dato da <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d33f5d498d528bd8c10edc8ac8c34347f32b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{1}}"></span> con autovalore 2. Per trovarne tutti gli autovalori si deve scrivere il polinomio caratteristico di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. Poiché la trasformazione è già scritta in forma di matrice, si procede con il calcolarne il polinomio caratteristico: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=\det(A-xI)={\begin{vmatrix}-x&1&-1\\1&1-x&0\\-1&0&1-x\end{vmatrix}}=-x^{3}+2x^{2}+x-2=-(x-2)(x-1)(x+1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>x</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>x</mi> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=\det(A-xI)={\begin{vmatrix}-x&1&-1\\1&1-x&0\\-1&0&1-x\end{vmatrix}}=-x^{3}+2x^{2}+x-2=-(x-2)(x-1)(x+1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/154eeb3472451df84b83b5c312cbd6dc68a0dfba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.089ex; width:92.041ex; height:9.176ex;" alt="{\displaystyle p(x)=\det(A-xI)={\begin{vmatrix}-x&1&-1\\1&1-x&0\\-1&0&1-x\end{vmatrix}}=-x^{3}+2x^{2}+x-2=-(x-2)(x-1)(x+1).}"></span></dd></dl> <p>Quindi gli autovalori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> sono 2, 1 e −1. I tre autovettori ortogonali sono: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}={\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}\qquad v_{2}={\begin{bmatrix}\;0\;\\1\\1\end{bmatrix}}\qquad v_{3}={\begin{bmatrix}\;2\\-1\\\;1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="2em" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>0</mn> <mspace width="thickmathspace" /> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="2em" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}={\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}\qquad v_{2}={\begin{bmatrix}\;0\;\\1\\1\end{bmatrix}}\qquad v_{3}={\begin{bmatrix}\;2\\-1\\\;1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456181e7fbb3e7c39ec9cd4c2b0d89255fa3a088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:45.728ex; height:9.176ex;" alt="{\displaystyle v_{1}={\begin{bmatrix}\;1\\\;1\\-1\end{bmatrix}}\qquad v_{2}={\begin{bmatrix}\;0\;\\1\\1\end{bmatrix}}\qquad v_{3}={\begin{bmatrix}\;2\\-1\\\;1\end{bmatrix}}.}"></span></dd></dl> <p>Per quanto detto prima, la trasformazione assume una forma molto semplice rispetto a questa base: ogni vettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> può essere scritto in modo unico come: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{1}v_{1}+x_{2}v_{2}+x_{3}v_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{1}v_{1}+x_{2}v_{2}+x_{3}v_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf43bbbe61903cd6ec06d0c8b52b44d20a6f9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.806ex; height:2.343ex;" alt="{\displaystyle x=x_{1}v_{1}+x_{2}v_{2}+x_{3}v_{3}}"></span></dd></dl> <p>e dunque si ha: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax=2x_{1}v_{1}+x_{2}v_{2}-x_{3}v_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax=2x_{1}v_{1}+x_{2}v_{2}-x_{3}v_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f00751b45c091cf19f269ed3b96b86eecead47e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.359ex; height:2.509ex;" alt="{\displaystyle Ax=2x_{1}v_{1}+x_{2}v_{2}-x_{3}v_{3}.}"></span></dd></dl> <p>Data infine una trasformazione lineare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, si è visto che se il polinomio caratteristico di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> ha tutte le radici in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span> con molteplicità 1, allora <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> è diagonalizzabile. Se invece il polinomio caratteristico di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> ha tutte le radici in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span> ma alcune di esse hanno molteplicità maggiore di 1, allora <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> non è necessariamente diagonalizzabile. Ad esempio la matrice: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left({\begin{matrix}1&0\\-a&1\end{matrix}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>a</mi> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left({\begin{matrix}1&0\\-a&1\end{matrix}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04dee973ec4430896dd2dd86f75e7b3323307986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.537ex; height:6.176ex;" alt="{\displaystyle A=\left({\begin{matrix}1&0\\-a&1\end{matrix}}\right)}"></span></dd></dl> <p>che rappresenta la trasformazione della <a href="/wiki/Monna_Lisa" class="mw-redirect" title="Monna Lisa">Gioconda</a> in figura ha come polinomio caratteristico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d33cfdca77932d4909be1ab8a0d25d8f22af594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.196ex; height:3.176ex;" alt="{\displaystyle (x-1)^{2}}"></span>, e non è diagonalizzabile per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Esempi_di_calcolo">Esempi di calcolo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=18" title="Modifica la sezione Esempi di calcolo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=18" title="Edit section's source code: Esempi di calcolo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Data la matrice di trasformazione: </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Eigenvectors.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/06/Eigenvectors.gif" decoding="async" width="300" height="300" class="mw-file-element" data-file-width="300" data-file-height="300" /></a><figcaption>Deformazione dello spazio bidimensionale a seguito della trasformazione operata dalla matrice A=<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl [}{\begin{smallmatrix}2&1\\1&2\end{smallmatrix}}{\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl [}{\begin{smallmatrix}2&1\\1&2\end{smallmatrix}}{\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcacd8bee0f5c4d9f5e8e3fc2c4932447e0e2aec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.919ex; margin-bottom: -0.253ex; width:5.108ex; height:3.343ex;" alt="{\displaystyle {\bigl [}{\begin{smallmatrix}2&1\\1&2\end{smallmatrix}}{\bigr ]}}"></span>. I vettori blu (che hanno la stessa direzione dell'autovettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle v_{1}={\bigl [}{\begin{smallmatrix}1\\1\end{smallmatrix}}{\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle v_{1}={\bigl [}{\begin{smallmatrix}1\\1\end{smallmatrix}}{\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd908778998b9669d6445cb12db4b536980773cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.033ex; height:3.509ex;" alt="{\displaystyle \scriptstyle v_{1}={\bigl [}{\begin{smallmatrix}1\\1\end{smallmatrix}}{\bigr ]}}"></span>) e i vettori viola (che hanno la stessa direzione dell'autovettore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle v_{2}={\bigl [}{\begin{smallmatrix}1\\-1\end{smallmatrix}}{\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle v_{2}={\bigl [}{\begin{smallmatrix}1\\-1\end{smallmatrix}}{\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a589d0af0fe7a4c60324da3be799b8a15aa856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.311ex; height:3.509ex;" alt="{\displaystyle \scriptstyle v_{2}={\bigl [}{\begin{smallmatrix}1\\-1\end{smallmatrix}}{\bigr ]}}"></span>) conservano la loro direzione anche dopo la trasformazione, a differenza dei vettori in rosso che sono orientati diversamente. Il quadrato iniziale a seguito della trasformazione si deforma diventando un rombo: i vettori blu triplicano il loro modulo avendo autovalore 3 mentre i viola restano inalterati avendo autovalore 1.</figcaption></figure> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}3&1\\1&3\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}3&1\\1&3\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d80070ef73ff16edc60561dfd7388b58522fbe13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.696ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}3&1\\1&3\end{bmatrix}}}"></span></dd></dl> <p>il vettore: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\begin{bmatrix}4\\-4\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\begin{bmatrix}4\\-4\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a111e575a5ea4f7139cc98f7b30b92ec18ea9586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.403ex; height:6.176ex;" alt="{\displaystyle v={\begin{bmatrix}4\\-4\end{bmatrix}}}"></span></dd></dl> <p>è un autovettore con autovalore 2. Infatti: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A*v={\begin{bmatrix}3&1\\1&3\end{bmatrix}}{\begin{bmatrix}4\\-4\end{bmatrix}}={\begin{bmatrix}3\cdot 4+1\cdot (-4)\\1\cdot 4+3\cdot (-4)\end{bmatrix}}={\begin{bmatrix}8\\-8\end{bmatrix}}=2\cdot {\begin{bmatrix}4\\-4\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∗</mo> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>4</mn> </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>3</mn> <mo>⋅</mo> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>⋅</mo> <mn>4</mn> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> </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>8</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>8</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A*v={\begin{bmatrix}3&1\\1&3\end{bmatrix}}{\begin{bmatrix}4\\-4\end{bmatrix}}={\begin{bmatrix}3\cdot 4+1\cdot (-4)\\1\cdot 4+3\cdot (-4)\end{bmatrix}}={\begin{bmatrix}8\\-8\end{bmatrix}}=2\cdot {\begin{bmatrix}4\\-4\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b3d03c92fe5e56cb865f22ce23e9d449f3a5d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.358ex; height:6.176ex;" alt="{\displaystyle A*v={\begin{bmatrix}3&1\\1&3\end{bmatrix}}{\begin{bmatrix}4\\-4\end{bmatrix}}={\begin{bmatrix}3\cdot 4+1\cdot (-4)\\1\cdot 4+3\cdot (-4)\end{bmatrix}}={\begin{bmatrix}8\\-8\end{bmatrix}}=2\cdot {\begin{bmatrix}4\\-4\end{bmatrix}}}"></span></dd></dl> <p>Per contro il vettore: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\begin{bmatrix}0\\1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\begin{bmatrix}0\\1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3172c7867bf88f52a3e51da20cbc924eba81e1ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.595ex; height:6.176ex;" alt="{\displaystyle v={\begin{bmatrix}0\\1\end{bmatrix}}}"></span></dd></dl> <p>non è un autovettore in quanto il vettore trasformato è: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}3&1\\1&3\end{bmatrix}}{\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}3\cdot 0+1\cdot 1\\1\cdot 0+3\cdot 1\end{bmatrix}}={\begin{bmatrix}1\\3\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </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>3</mn> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mn>1</mn> </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> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}3&1\\1&3\end{bmatrix}}{\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}3\cdot 0+1\cdot 1\\1\cdot 0+3\cdot 1\end{bmatrix}}={\begin{bmatrix}1\\3\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0972dada969669b1432eff43a69d5f3448fa7694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.844ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}3&1\\1&3\end{bmatrix}}{\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}3\cdot 0+1\cdot 1\\1\cdot 0+3\cdot 1\end{bmatrix}}={\begin{bmatrix}1\\3\end{bmatrix}}}"></span></dd></dl> <p>e, come si nota facilmente, manca la proporzionalità tra il vettore trasformato <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl [}{\begin{smallmatrix}1\\3\end{smallmatrix}}{\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl [}{\begin{smallmatrix}1\\3\end{smallmatrix}}{\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc342fe475c1f3e16d616bda8753bd64e6c9e34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.513ex; height:3.343ex;" alt="{\displaystyle {\bigl [}{\begin{smallmatrix}1\\3\end{smallmatrix}}{\bigr ]}}"></span> e il vettore originale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl [}{\begin{smallmatrix}0\\1\end{smallmatrix}}{\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl [}{\begin{smallmatrix}0\\1\end{smallmatrix}}{\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eacfd7e23aaf01c0ae72792d11542f24451804c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.513ex; height:3.343ex;" alt="{\displaystyle {\bigl [}{\begin{smallmatrix}0\\1\end{smallmatrix}}{\bigr ]}}"></span>, condizione necessaria per il parallelismo. </p><p>Come secondo esempio, si consideri la matrice di trasformazione: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef64681325a9c3a34251382c5c48875827aba7fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.696ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}}"></span></dd></dl> <p>è facile verificare che i vettori: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1}={\begin{bmatrix}1\\1\end{bmatrix}}\qquad v_{2}={\begin{bmatrix}1\\-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="2em" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}={\begin{bmatrix}1\\1\end{bmatrix}}\qquad v_{2}={\begin{bmatrix}1\\-1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b63f0c8aaa1c97487f917157e482702697682aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.752ex; height:6.176ex;" alt="{\displaystyle v_{1}={\begin{bmatrix}1\\1\end{bmatrix}}\qquad v_{2}={\begin{bmatrix}1\\-1\end{bmatrix}}}"></span></dd></dl> <p>sono autovettori con autovalori 3 e 1 rispettivamente. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=19" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=19" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-def-1"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-def_1-0">a</a></i></sup> <sup><i><a href="#cite_ref-def_1-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 220</cite>.</span> </li> <li id="cite_note-autospazio-2"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-autospazio_2-0">a</a></i></sup> <sup><i><a href="#cite_ref-autospazio_2-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 221</cite>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 104</cite>.</span> </li> <li id="cite_note-4"><a href="#cite_ref-4"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 105</cite>.</span> </li> <li id="cite_note-5"><a href="#cite_ref-5"><b>^</b></a> <span class="reference-text"><a rel="nofollow" class="external text" href="http://web.stanford.edu/class/cme335/lecture4sup.pdf">Jim Lambers - The Unsymmetric Eigenvalue Problem</a></span> </li> <li id="cite_note-6"><a href="#cite_ref-6"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 227</cite>.</span> </li> <li id="cite_note-7"><a href="#cite_ref-7"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 228</cite>.</span> </li> <li id="cite_note-8"><a href="#cite_ref-8"><b>^</b></a> <span class="reference-text">Nella pratica gli autovalori di grandi matrici non vengono calcolati usando il polinomio caratteristico, esistendo metodi <a href="/wiki/Analisi_numerica" title="Analisi numerica">numerici</a> più veloci e sufficientemente stabili.</span> </li> <li id="cite_note-9"><a href="#cite_ref-9"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 223</cite>.</span> </li> <li id="cite_note-10"><a href="#cite_ref-10"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 114</cite>.</span> </li> <li id="cite_note-11"><a href="#cite_ref-11"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 115</cite>.</span> </li> <li id="cite_note-12"><a href="#cite_ref-12"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFlang">S. Lang</a>, Pag. 251</cite>.</span> </li> <li id="cite_note-13"><a href="#cite_ref-13"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFreed">Reed, Simon</a>, Pag. 188</cite>.</span> </li> <li id="cite_note-14"><a href="#cite_ref-14"><b>^</b></a> <span class="reference-text">Lo <a href="/wiki/Shift_unilaterale" class="mw-redirect" title="Shift unilaterale">shift unilaterale</a> su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l^{2}(N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l^{2}(N)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e31c39b3b0d560871cabe59a0076fd47002d60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.62ex; height:3.176ex;" alt="{\displaystyle l^{2}(N)}"></span> ne fornisce un esempio: tale operatore è una <a href="/wiki/Isometria" title="Isometria">isometria</a>, ed è quindi limitato ma non invertibile poiché non è surriettivo.</span> </li> <li id="cite_note-15"><a href="#cite_ref-15"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFreed">Reed, Simon</a>, Pag. 194</cite>.</span> </li> <li id="cite_note-16"><a href="#cite_ref-16"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a href="/w/index.php?title=John_Keating&action=edit&redlink=1" class="new" title="John Keating (la pagina non esiste)">John Keating</a>, <span style="font-style:italic;">L-functions and the Characteristic Polynomials of Random Matrices</span>, in <a href="/wiki/Francesco_Mezzadri" title="Francesco Mezzadri">Francesco Mezzadri</a> e <a href="/w/index.php?title=Nina_Snaith&action=edit&redlink=1" class="new" title="Nina Snaith (la pagina non esiste)">Nina Snaith</a> (a cura di), <span style="font-style:italic;">Recent perspectives in random matrix theory and number theory</span>, Cambridge, Cambridge University Press, 2005, pp. 251-278, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-521-62058-1" title="Speciale:RicercaISBN/978-0-521-62058-1">978-0-521-62058-1</a>.</cite></span> </li> <li id="cite_note-17"><a href="#cite_ref-17"><b>^</b></a> <span class="reference-text"><cite class="citation pubblicazione" style="font-style:normal"> A. Xirouhakis, G. Votsis e A. Delopoulus, <a rel="nofollow" class="external text" href="http://www.image.ece.ntua.gr/papers/43.pdf"><span style="font-style:italic;">Estimation of 3D motion and structure of human faces</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="documento in formato PDF">PDF</abbr></span>), Online paper in PDF format, National Technical University of Athens, 2004.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=20" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=20" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFlang" class="citation libro" style="font-style:normal"> Serge Lang, <span style="font-style:italic;">Algebra lineare</span>, Torino, <a href="/wiki/Bollati_Boringhieri" title="Bollati Boringhieri">Bollati Boringhieri</a>, 1992, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-339-5035-2" title="Speciale:RicercaISBN/88-339-5035-2">88-339-5035-2</a>.</cite></li> <li><cite id="CITEREFreed" class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Michael Reed, Barry Simon, <span style="font-style:italic;">Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis</span>, 2ª ed., San Diego, California, Academic press inc., 1980, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-12-585050-6" title="Speciale:RicercaISBN/0-12-585050-6">0-12-585050-6</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Marius Stoka, <span style="font-style:italic;">Corso di geometria</span>, Cedam, 1995, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-13-19192-4" title="Speciale:RicercaISBN/978-88-13-19192-4">978-88-13-19192-4</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a>, <span style="font-style:italic;">Algebra, 3rd edition</span>, <a href="/wiki/Springer_(azienda)" title="Springer (azienda)">Springer</a>, 2002, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-387-95385-X" title="Speciale:RicercaISBN/0-387-95385-X">0-387-95385-X</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Steven Roman, <a rel="nofollow" class="external text" href="https://archive.org/details/springer_10.1007-978-1-4757-2178-2"><span style="font-style:italic;">Advanced Linear Algebra</span></a>, Springer, 1992, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-387-97837-2" title="Speciale:RicercaISBN/0-387-97837-2">0-387-97837-2</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a href="/wiki/Paul_Richard_Halmos" class="mw-redirect" title="Paul Richard Halmos">Paul R. Halmos</a>, <span style="font-style:italic;">Finite-dimensional Vector Spaces</span>, Springer, 1993, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-387-90093-4" title="Speciale:RicercaISBN/0-387-90093-4">0-387-90093-4</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Werner H. Greub, <span style="font-style:italic;">Linear Algebra, 4th ed.</span>, Springer, 1981, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-387-90110-8" title="Speciale:RicercaISBN/0-387-90110-8">0-387-90110-8</a>.</cite></li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Jim Hefferon, <i>Linear Algebra</i>, <a rel="nofollow" class="external text" href="http://joshua.smcvt.edu/linearalgebra/">Online book</a>, St Michael's College, Colchester, Vermont, USA 2001.</li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Gene H. Golub e Charles F. van Loan, <span style="font-style:italic;">Matrix computations, 3rd Edition</span>, Johns Hopkins University Press, 1996, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-8018-5414-8" title="Speciale:RicercaISBN/0-8018-5414-8">0-8018-5414-8</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a href="/w/index.php?title=Nelson_Dunford&action=edit&redlink=1" class="new" title="Nelson Dunford (la pagina non esiste)">Nelson Dunford</a> e <a href="/wiki/Jacob_Schwartz" title="Jacob Schwartz">Jacob Schwartz</a>, <span style="font-style:italic;">Linear Operator. Part I General Theory</span>, Wiley-Interscience, 1958, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-471-60848-3" title="Speciale:RicercaISBN/0-471-60848-3">0-471-60848-3</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) V. G. Prikazchikov, <span style="font-style:italic;">Eigen values of differential operators, numerical methods</span>.</cite> <a rel="nofollow" class="external text" href="http://eom.springer.de/E/e035160.htm">accessibile</a> in <a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) A. B. Bakushinskii, <span style="font-style:italic;">Eigen values of integral operators, numerical methods</span>.</cite><a rel="nofollow" class="external text" href="http://eom.springer.de/E/e035170.htm">accessibile</a> in <a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a href="/wiki/Leonid_Vital%27evi%C4%8D_Kantorovi%C4%8D" title="Leonid Vital'evič Kantorovič">Leonid V. Kantorovič</a> e G. P. Akilov, <span style="font-style:italic;">Functional analysis</span>, Pergamon Press, 1982, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-14-83-11532-0" title="Speciale:RicercaISBN/978-14-83-11532-0">978-14-83-11532-0</a>.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=21" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=21" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="colonne"> <ul><li><a href="/wiki/Autofunzione" title="Autofunzione">Autofunzione</a></li> <li><a href="/wiki/Autostato" title="Autostato">Autostato</a></li> <li><a href="/wiki/Decomposizione_ai_valori_singolari" title="Decomposizione ai valori singolari">Decomposizione ai valori singolari</a></li> <li><a href="/wiki/Decomposizione_di_Schur" title="Decomposizione di Schur">Decomposizione di Schur</a></li> <li><a href="/wiki/Diagonalizzabilit%C3%A0" title="Diagonalizzabilità">Diagonalizzabilità</a></li> <li><a href="/wiki/Forma_canonica_di_Jordan" title="Forma canonica di Jordan">Forma canonica di Jordan</a></li> <li><a href="/wiki/Polinomio_caratteristico" title="Polinomio caratteristico">Polinomio caratteristico</a></li> <li><a href="/wiki/Raggio_spettrale" title="Raggio spettrale">Raggio spettrale</a></li> <li><a href="/wiki/Similitudine_fra_matrici" class="mw-redirect" title="Similitudine fra matrici">Similitudine fra matrici</a></li> <li><a href="/wiki/Spettro_(matematica)" title="Spettro (matematica)">Spettro (matematica)</a></li> <li><a href="/wiki/Teorema_spettrale" title="Teorema spettrale">Teorema spettrale</a></li> <li><a href="/wiki/Teoremi_di_Gerschgorin" class="mw-redirect" title="Teoremi di Gerschgorin">Teoremi di Gerschgorin</a></li> <li><a href="/wiki/Teorema_di_Schur-Horn" title="Teorema di Schur-Horn">Teorema di Schur-Horn</a></li> <li><a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">Trasformazione lineare</a></li> <li><a href="/wiki/Valore_singolare" title="Valore singolare">Valore singolare</a></li> <li><a href="/wiki/Metodo_delle_potenze_inverse" title="Metodo delle potenze inverse">Metodo delle potenze inverse</a>: metodo numerico per trovare gli autovalori</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=22" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=22" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wiktionary.org/wiki/autovettore" class="extiw" title="wikt:autovettore">Wikizionario</a></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a href="https://it.wiktionary.org/wiki/autovettore" class="extiw" title="wikt:autovettore">autovettore</a></b>»</li></ul> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wiktionary.org/wiki/autovalore" class="extiw" title="wikt:autovalore">Wikizionario</a></li> <li class="" title=""><a href="https://it.wikiversity.org/wiki/Autovalori_e_autovettori_e_diagonalizzazione_di_matrici" class="extiw" title="v:Autovalori e autovettori e diagonalizzazione di matrici">Wikiversità</a></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a href="https://it.wiktionary.org/wiki/autovalore" class="extiw" title="wikt:autovalore">autovalore</a></b>»</li> <li><span typeof="mw:File"><a href="https://it.wikiversity.org/wiki/" title="Collabora a Wikiversità"><img alt="Collabora a Wikiversità" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/18px-Wikiversity_logo_2017.svg.png" decoding="async" width="18" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/36px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span> <a href="https://it.wikiversity.org/wiki/" class="extiw" title="v:">Wikiversità</a> contiene risorse su <b><a href="https://it.wikiversity.org/wiki/Autovalori_e_autovettori_e_diagonalizzazione_di_matrici" class="extiw" title="v:Autovalori e autovettori e diagonalizzazione di matrici">autovettore e autovalore</a></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Autovettore_e_autovalore&veaction=edit&section=23" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Autovettore_e_autovalore&action=edit&section=23" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFEnciclopedia_della_Matematica" class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/autovalore_(Enciclopedia-della-Matematica)/"><span style="font-style:italic;">autovalore</span></a>, in <span style="font-style:italic;">Enciclopedia della Matematica</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2013.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q190524#P9621" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFSpringerEOM" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Eigen_vector"><span style="font-style:italic;">Eigen vector</span></a> / <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Root_vector"><span style="font-style:italic;">Root vector</span></a>, su <span style="font-style:italic;"><a href="/wiki/Encyclopaedia_of_Mathematics" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></span>, Springer e European Mathematical Society.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q190524#P7554" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li></ul> <dl><dt>Calcolatrici online</dt></dl> <ul><li><a rel="nofollow" class="external text" href="http://www.arndt-bruenner.de/mathe/scripts/engl_eigenwert.htm">Calculator for Eigenvalues</a> nel sito di Arndt Brünner</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081212221215/http://www.bluebit.gr/matrix-calculator/">Online Matrix Calculator</a> presso BlueBit Software</li> <li><a rel="nofollow" class="external text" href="http://wims.unice.fr/wims/wims.cgi?session=6S051ABAFA.2&+lang=en&+module=tool%2Flinear%2Fmatrix.en">Matrix calculator</a> in WIMS, WWW Interactive Multipurpose Server, presso l'Université Nice Sophia Antipolis</li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Algebra_lineare"><tbody><tr><th colspan="3" style="background:#99CCFF"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Algebra_lineare" title="Template:Algebra lineare"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Algebra_lineare&action=edit&redlink=1" class="new" title="Discussioni template:Algebra lineare (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Algebra_lineare&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Algebra_lineare" title="Algebra lineare">Algebra lineare</a></span></th></tr><tr><th colspan="1" class="navbox_group" style="background:#fff; text-align:right;"><a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">Spazio vettoriale</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">Vettore</a><b> ·</b> <a href="/wiki/Sottospazio_vettoriale" title="Sottospazio vettoriale">Sottospazio vettoriale</a> <small>(<a href="/wiki/Copertura_lineare" title="Copertura lineare">Sottospazio generato</a>)</small><b> ·</b> <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">Applicazione lineare</a> <small>(<a href="/wiki/Nucleo_(matematica)" title="Nucleo (matematica)">Nucleo</a><b> ·</b> <a href="/wiki/Immagine_(matematica)" title="Immagine (matematica)">Immagine</a>)</small><b> ·</b> <a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">Base</a><b> ·</b> <a href="/wiki/Dimensione_(spazio_vettoriale)" title="Dimensione (spazio vettoriale)">Dimensione</a><b> ·</b> <a href="/wiki/Teorema_del_rango" title="Teorema del rango">Teorema della dimensione</a><b> ·</b> <a href="/wiki/Formula_di_Grassmann" title="Formula di Grassmann">Formula di Grassmann</a><b> ·</b> <a href="/wiki/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari">Sistema lineare</a><b> ·</b> <a href="/wiki/Metodo_di_eliminazione_di_Gauss" title="Metodo di eliminazione di Gauss">Algoritmo di Gauss</a><b> ·</b> <a href="/wiki/Teorema_di_Rouch%C3%A9-Capelli" title="Teorema di Rouché-Capelli">Teorema di Rouché-Capelli</a><b> ·</b> <a href="/wiki/Regola_di_Cramer" title="Regola di Cramer">Regola di Cramer</a><b> ·</b> <a href="/wiki/Spazio_duale" title="Spazio duale">Spazio duale</a><b> ·</b> <a href="/wiki/Spazio_proiettivo" title="Spazio proiettivo">Spazio proiettivo</a><b> ·</b> <a href="/wiki/Spazio_affine" title="Spazio affine">Spazio affine</a><b> ·</b> <a href="/wiki/Teorema_della_dimensione_per_spazi_vettoriali" title="Teorema della dimensione per spazi vettoriali">Teorema della dimensione per spazi vettoriali</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#fff; text-align:right;"><a href="/wiki/Matrice" title="Matrice">Matrici</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Matrice_identit%C3%A0" title="Matrice identità">Identità</a><b> ·</b> <a href="/wiki/Matrice_nulla" title="Matrice nulla">Nulla</a><b> ·</b> <a href="/wiki/Matrice_quadrata" title="Matrice quadrata">Quadrata</a><b> ·</b> <a href="/wiki/Matrice_invertibile" title="Matrice invertibile">Invertibile</a><b> ·</b> <a href="/wiki/Matrice_simmetrica" title="Matrice simmetrica">Simmetrica</a><b> ·</b> <a href="/wiki/Matrice_antisimmetrica" title="Matrice antisimmetrica">Antisimmetrica</a><b> ·</b> <a href="/wiki/Matrice_trasposta" title="Matrice trasposta">Trasposta</a><b> ·</b> <a href="/wiki/Matrice_diagonale" title="Matrice diagonale">Diagonale</a><b> ·</b> <a href="/wiki/Matrice_triangolare" title="Matrice triangolare">Triangolare</a><b> ·</b> <a href="/wiki/Matrice_di_cambiamento_di_base" title="Matrice di cambiamento di base">Di cambiamento di base</a><b> ·</b> <a href="/wiki/Matrice_ortogonale" title="Matrice ortogonale">Ortogonale</a><b> ·</b> <a href="/wiki/Matrice_normale" title="Matrice normale">Normale</a><b> ·</b> <a href="/wiki/Matrice_di_rotazione" title="Matrice di rotazione">Rotazione</a><b> ·</b> <a href="/wiki/Matrice_simplettica" title="Matrice simplettica">Simplettica</a><b> ·</b> <a href="/wiki/Moltiplicazione_di_matrici" title="Moltiplicazione di matrici">Moltiplicazione di matrici</a><b> ·</b> <a href="/wiki/Rango_(algebra_lineare)" title="Rango (algebra lineare)">Rango</a><b> ·</b> <a href="/wiki/Teorema_di_Kronecker" title="Teorema di Kronecker">Teorema di Kronecker</a><b> ·</b> <a href="/wiki/Minore_(algebra_lineare)" title="Minore (algebra lineare)">Minore</a><b> ·</b> <a href="/wiki/Matrice_dei_cofattori" title="Matrice dei cofattori">Matrice dei cofattori</a><b> ·</b> <a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">Determinante</a><b> ·</b> <a href="/wiki/Teorema_di_Binet" title="Teorema di Binet">Teorema di Binet</a><b> ·</b> <a href="/wiki/Teorema_di_Laplace" title="Teorema di Laplace">Teorema di Laplace</a><b> ·</b> <a href="/wiki/Radice_quadrata_di_una_matrice" title="Radice quadrata di una matrice">Radice quadrata di una matrice</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#fff; text-align:right;"><a href="/wiki/Diagonalizzabilit%C3%A0" title="Diagonalizzabilità">Diagonalizzabilità</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a class="mw-selflink selflink">Autovettore e autovalore</a><b> ·</b> <a href="/wiki/Spettro_(matematica)" title="Spettro (matematica)">Spettro</a><b> ·</b> <a href="/wiki/Polinomio_caratteristico" title="Polinomio caratteristico">Polinomio caratteristico</a><b> ·</b> <a href="/wiki/Polinomio_minimo" title="Polinomio minimo">Polinomio minimo</a><b> ·</b> <a href="/wiki/Teorema_di_Hamilton-Cayley" title="Teorema di Hamilton-Cayley">Teorema di Hamilton-Cayley</a><b> ·</b> <a href="/wiki/Matrice_a_blocchi" title="Matrice a blocchi">Matrice a blocchi</a><b> ·</b> <a href="/wiki/Forma_canonica_di_Jordan" title="Forma canonica di Jordan">Forma canonica di Jordan</a><b> ·</b> <a href="/wiki/Teorema_di_diagonalizzabilit%C3%A0" title="Teorema di diagonalizzabilità">Teorema di diagonalizzabilità</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#fff; 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Autovettore e autovalore - Wikipedia