Matrice - Wikipedia

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class="vector-toc-link" href="#Definizioni_e_notazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definizioni e notazioni</span> </div> </a> <ul id="toc-Definizioni_e_notazioni-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebra_delle_matrici" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebra_delle_matrici"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Algebra delle matrici</span> </div> </a> <button aria-controls="toc-Algebra_delle_matrici-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 Algebra delle matrici</span> </button> <ul id="toc-Algebra_delle_matrici-sublist" class="vector-toc-list"> <li id="toc-Somma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Somma"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Somma</span> </div> </a> <ul id="toc-Somma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Moltiplicazione_per_uno_scalare" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Moltiplicazione_per_uno_scalare"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Moltiplicazione per uno scalare</span> </div> </a> <ul id="toc-Moltiplicazione_per_uno_scalare-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prodotto" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prodotto"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Prodotto</span> </div> </a> <ul id="toc-Prodotto-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proprietà" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proprietà"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Proprietà</span> </div> </a> <ul id="toc-Proprietà-sublist" class="vector-toc-list"> <li id="toc-Proprietà_della_somma_e_del_prodotto_per_uno_scalare" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Proprietà_della_somma_e_del_prodotto_per_uno_scalare"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.1</span> <span>Proprietà della somma e del prodotto per uno scalare</span> </div> </a> <ul id="toc-Proprietà_della_somma_e_del_prodotto_per_uno_scalare-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proprietà_del_prodotto_fra_matrici" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Proprietà_del_prodotto_fra_matrici"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.2</span> <span>Proprietà del prodotto fra matrici</span> </div> </a> <ul id="toc-Proprietà_del_prodotto_fra_matrici-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Altre_operazioni" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Altre_operazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Altre operazioni</span> </div> </a> <ul id="toc-Altre_operazioni-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applicazioni_lineari" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applicazioni_lineari"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applicazioni lineari</span> </div> </a> <button aria-controls="toc-Applicazioni_lineari-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 lineari</span> </button> <ul id="toc-Applicazioni_lineari-sublist" class="vector-toc-list"> <li id="toc-Sistemi_lineari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sistemi_lineari"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Sistemi lineari</span> </div> </a> <ul id="toc-Sistemi_lineari-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Matrici_quadrate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrici_quadrate"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Matrici quadrate</span> </div> </a> <button aria-controls="toc-Matrici_quadrate-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 Matrici quadrate</span> </button> <ul id="toc-Matrici_quadrate-sublist" class="vector-toc-list"> <li id="toc-Prodotto_di_matrici_quadrate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prodotto_di_matrici_quadrate"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Prodotto di matrici quadrate</span> </div> </a> <ul id="toc-Prodotto_di_matrici_quadrate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Determinante" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Determinante"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Determinante</span> </div> </a> <ul id="toc-Determinante-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polinomio_caratteristico,_autovettori,_diagonalizzabilità" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polinomio_caratteristico,_autovettori,_diagonalizzabilità"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Polinomio caratteristico, autovettori, diagonalizzabilità</span> </div> </a> <ul id="toc-Polinomio_caratteristico,_autovettori,_diagonalizzabilità-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Classi_di_matrici_reali_e_complesse" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classi_di_matrici_reali_e_complesse"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Classi di matrici reali e complesse</span> </div> </a> <ul id="toc-Classi_di_matrici_reali_e_complesse-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spazio_di_matrici" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spazio_di_matrici"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Spazio di matrici</span> </div> </a> <button aria-controls="toc-Spazio_di_matrici-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 Spazio di matrici</span> </button> <ul id="toc-Spazio_di_matrici-sublist" class="vector-toc-list"> <li id="toc-Algebra_su_campo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebra_su_campo"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Algebra su campo</span> </div> </a> <ul id="toc-Algebra_su_campo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizzazioni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizzazioni"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Generalizzazioni</span> </div> </a> <ul id="toc-Generalizzazioni-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funzione_di_matrice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funzione_di_matrice"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Funzione di matrice</span> </div> </a> <ul id="toc-Funzione_di_matrice-sublist" class="vector-toc-list"> </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">10</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">11</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">12</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">13</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">14</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">Matrice</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 92 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-92" 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">92 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Matriks" title="Matriks - afrikaans" lang="af" hreflang="af" data-title="Matriks" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%9B%E1%89%B5%E1%88%AA%E1%8A%AD%E1%88%B5" title="ማትሪክስ - amarico" lang="am" hreflang="am" data-title="ማትሪክስ" data-language-autonym="አማርኛ" data-language-local-name="amarico" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مصفوفة (رياضيات) - arabo" lang="ar" hreflang="ar" data-title="مصفوفة (رياضيات)" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%D9%8A%D8%B3" title="ماتريس - arabo marocchino" lang="ary" hreflang="ary" data-title="ماتريس" data-language-autonym="الدارجة" data-language-local-name="arabo marocchino" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Matris" title="Matris - azerbaigiano" lang="az" hreflang="az" data-title="Matris" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaigiano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Матрыца (матэматыка) - bielorusso" lang="be" hreflang="be" data-title="Матрыца (матэматыка)" 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%9C%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0" title="Матрыца - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матрыца" 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%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) - bulgaro" lang="bg" hreflang="bg" data-title="Матрица (математика)" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%9F%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A7%8D%E0%A6%B8" title="ম্যাট্রিক্স - bengalese" lang="bn" hreflang="bn" data-title="ম্যাট্রিক্স" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) - bosniaco" lang="bs" hreflang="bs" data-title="Matrica (matematika)" data-language-autonym="Bosanski" data-language-local-name="bosniaco" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matriu_(matem%C3%A0tiques)" title="Matriu (matemàtiques) - catalano" lang="ca" hreflang="ca" data-title="Matriu (matemàtiques)" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%DA%A9%D8%B3" title="ماتریکس - curdo centrale" lang="ckb" hreflang="ckb" data-title="ماتریکس" data-language-autonym="کوردی" data-language-local-name="curdo centrale" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matice" title="Matice - ceco" lang="cs" hreflang="cs" data-title="Matice" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) - ciuvascio" lang="cv" hreflang="cv" data-title="Матрица (математика)" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Matrics" title="Matrics - gallese" lang="cy" hreflang="cy" data-title="Matrics" data-language-autonym="Cymraeg" data-language-local-name="gallese" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Matrix" title="Matrix - danese" lang="da" hreflang="da" data-title="Matrix" 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/Matrix_(Mathematik)" title="Matrix (Mathematik) - tedesco" lang="de" hreflang="de" data-title="Matrix (Mathematik)" 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%A0%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Πίνακας (μαθηματικά) - greco" lang="el" hreflang="el" data-title="Πίνακας (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="voce di qualità"><a href="https://en.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics) - inglese" lang="en" hreflang="en" data-title="Matrix (mathematics)" 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/Matrico" title="Matrico - esperanto" lang="eo" hreflang="eo" data-title="Matrico" 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/Matriz_(matem%C3%A1tica)" title="Matriz (matemática) - spagnolo" lang="es" hreflang="es" data-title="Matriz (matemática)" 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/Maatriks" title="Maatriks - estone" lang="et" hreflang="et" data-title="Maatriks" 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/Matrize" title="Matrize - basco" lang="eu" hreflang="eu" data-title="Matrize" 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%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3" title="ماتریس - persiano" lang="fa" hreflang="fa" data-title="ماتریس" 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/Matriisi" title="Matriisi - finlandese" lang="fi" hreflang="fi" data-title="Matriisi" 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/Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques) - francese" lang="fr" hreflang="fr" data-title="Matrice (mathématiques)" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Maatriks" title="Maatriks - frisone settentrionale" lang="frr" hreflang="frr" data-title="Maatriks" data-language-autonym="Nordfriisk" data-language-local-name="frisone settentrionale" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Maitr%C3%ADs" title="Maitrís - irlandese" lang="ga" hreflang="ga" data-title="Maitrís" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E8%A1%8C%E5%88%97" title="行列 - gan" lang="gan" hreflang="gan" data-title="行列" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Matris_(mat%C3%A9matik)" title="Matris (matématik) - Guianan Creole" lang="gcr" hreflang="gcr" data-title="Matris (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Matriz_(matem%C3%A1ticas)" title="Matriz (matemáticas) - galiziano" lang="gl" hreflang="gl" data-title="Matriz (matemáticas)" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94" title="מטריצה - ebraico" lang="he" hreflang="he" data-title="מטריצה" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%B5%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%B9" title="आव्यूह - hindi" lang="hi" hreflang="hi" data-title="आव्यूह" 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/Matrica_(matematika)" title="Matrica (matematika) - croato" lang="hr" hreflang="hr" data-title="Matrica (matematika)" 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/M%C3%A1trix_(matematika)" title="Mátrix (matematika) - ungherese" lang="hu" hreflang="hu" data-title="Mátrix (matematika)" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%BF%D6%80%D5%AB%D6%81" title="Մատրից - armeno" lang="hy" hreflang="hy" data-title="Մատրից" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Matrice_(mathematica)" title="Matrice (mathematica) - interlingua" lang="ia" hreflang="ia" data-title="Matrice (mathematica)" 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/Matriks_(matematika)" title="Matriks (matematika) - indonesiano" lang="id" hreflang="id" data-title="Matriks (matematika)" 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/Fylki_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Fylki (stærðfræði) - islandese" lang="is" hreflang="is" data-title="Fylki (stærðfræði)" 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/%E8%A1%8C%E5%88%97" title="行列 - giapponese" lang="ja" hreflang="ja" data-title="行列" 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-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%A2%E1%83%A0%E1%83%98%E1%83%AA%E1%83%90_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="მატრიცა (მათემატიკა) - georgiano" lang="ka" hreflang="ka" data-title="მატრიცა (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) - kazako" lang="kk" hreflang="kk" data-title="Матрица (математика)" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AE%E0%B2%BE%E0%B2%A4%E0%B3%83%E0%B2%95%E0%B3%86%E0%B2%97%E0%B2%B3%E0%B3%81" title="ಮಾತೃಕೆಗಳು - kannada" lang="kn" hreflang="kn" data-title="ಮಾತೃಕೆಗಳು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%96%89%EB%A0%AC" title="행렬 - coreano" lang="ko" hreflang="ko" data-title="행렬" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Matrix_(mathematica)" title="Matrix (mathematica) - latino" lang="la" hreflang="la" data-title="Matrix (mathematica)" data-language-autonym="Latina" data-language-local-name="latino" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Matris" title="Matris - lombardo" lang="lmo" hreflang="lmo" data-title="Matris" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%A1%E0%BA%B2%E0%BA%95%E0%BA%A3%E0%BA%B4%E0%BA%81" title="ມາຕຣິກ - lao" lang="lo" hreflang="lo" data-title="ມາຕຣິກ" data-language-autonym="ລາວ" data-language-local-name="lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) - lituano" lang="lt" hreflang="lt" data-title="Matrica (matematika)" 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/Matrica" title="Matrica - lettone" lang="lv" hreflang="lv" data-title="Matrica" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B5" title="Матрице - Eastern Mari" lang="mhr" hreflang="mhr" data-title="Матрице" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) - macedone" lang="mk" hreflang="mk" data-title="Матрица (математика)" data-language-autonym="Македонски" data-language-local-name="macedone" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AE%E0%B4%BE%E0%B4%9F%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B5%8D%E0%B4%B8%E0%B5%8D" title="മാട്രിക്സ് - malayalam" lang="ml" hreflang="ml" data-title="മാട്രിക്സ്" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Matriks_(matematik)" title="Matriks (matematik) - malese" lang="ms" hreflang="ms" data-title="Matriks (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%A1%E1%80%AF%E1%80%B6" title="ကိန်းအုံ - birmano" lang="my" hreflang="my" data-title="ကိန်းအုံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmano" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AE%E0%A5%87%E0%A4%9F%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8" title="मेट्रिक्स - nepalese" lang="ne" hreflang="ne" data-title="मेट्रिक्स" data-language-autonym="नेपाली" data-language-local-name="nepalese" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Matrix_(wiskunde)" title="Matrix (wiskunde) - olandese" lang="nl" hreflang="nl" data-title="Matrix (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matrise" title="Matrise - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Matrise" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matrise" title="Matrise - norvegese bokmål" lang="nb" hreflang="nb" data-title="Matrise" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Tareentaa_(Maatiriksii)" title="Tareentaa (Maatiriksii) - oromo" lang="om" hreflang="om" data-title="Tareentaa (Maatiriksii)" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AE%E0%AC%BE%E0%AC%9F%E0%AD%8D%E0%AC%B0%E0%AC%BF%E0%AC%95%E0%AD%8D%E0%AC%B8" title="ମାଟ୍ରିକ୍ସ - odia" lang="or" hreflang="or" data-title="ମାଟ୍ରିକ୍ସ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AE%E0%A9%88%E0%A8%9F%E0%A9%8D%E0%A8%B0%E0%A8%BF%E0%A8%95%E0%A8%B8_(%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4)" title="ਮੈਟ੍ਰਿਕਸ (ਗਣਿਤ) - punjabi" lang="pa" hreflang="pa" data-title="ਮੈਟ੍ਰਿਕਸ (ਗਣਿਤ)" 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/Macierz" title="Macierz - polacco" lang="pl" hreflang="pl" data-title="Macierz" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Matris" title="Matris - piemontese" lang="pms" hreflang="pms" data-title="Matris" data-language-autonym="Piemontèis" data-language-local-name="piemontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D8%A7%D9%B9%D8%B1%DA%A9%D8%B3_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="ماٹرکس (ریاضیات) - Western Punjabi" lang="pnb" hreflang="pnb" data-title="ماٹرکس (ریاضیات)" 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/Matriz_(matem%C3%A1tica)" title="Matriz (matemática) - portoghese" lang="pt" hreflang="pt" data-title="Matriz (matemática)" 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/Matrice" title="Matrice - rumeno" lang="ro" hreflang="ro" data-title="Matrice" data-language-autonym="Română" data-language-local-name="rumeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) - russo" lang="ru" hreflang="ru" data-title="Матрица (математика)" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) - sacha" lang="sah" hreflang="sah" data-title="Матрица (математика)" data-language-autonym="Саха тыла" data-language-local-name="sacha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Matrici_(matim%C3%A0tica)" title="Matrici (matimàtica) - siciliano" lang="scn" hreflang="scn" data-title="Matrici (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics) - scozzese" lang="sco" hreflang="sco" data-title="Matrix (mathematics)" data-language-autonym="Scots" data-language-local-name="scozzese" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) - serbo-croato" lang="sh" hreflang="sh" data-title="Matrica (matematika)" 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-si badge-Q17437798 badge-goodarticle mw-list-item" title="voce di qualità"><a href="https://si.wikipedia.org/wiki/%E0%B6%B1%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%83_(%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%BA)" title="න්‍යාස (ගණිතය) - singalese" lang="si" hreflang="si" data-title="න්‍යාස (ගණිතය)" data-language-autonym="සිංහල" data-language-local-name="singalese" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics) - Simple English" lang="en-simple" hreflang="en-simple" data-title="Matrix (mathematics)" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matica_(matematika)" title="Matica (matematika) - slovacco" lang="sk" hreflang="sk" data-title="Matica (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovacco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matrika" title="Matrika - sloveno" lang="sl" hreflang="sl" data-title="Matrika" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Taxane" title="Taxane - somalo" lang="so" hreflang="so" data-title="Taxane" data-language-autonym="Soomaaliga" data-language-local-name="somalo" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matrica" title="Matrica - albanese" lang="sq" hreflang="sq" data-title="Matrica" 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/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) - serbo" lang="sr" hreflang="sr" data-title="Матрица (математика)" 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/Matris" title="Matris - svedese" lang="sv" hreflang="sv" data-title="Matris" 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%85%E0%AE%A3%E0%AE%BF_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="அணி (கணிதம்) - tamil" lang="ta" hreflang="ta" data-title="அணி (கணிதம்)" 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%B9%80%E0%B8%A1%E0%B8%97%E0%B8%A3%E0%B8%B4%E0%B8%81%E0%B8%8B%E0%B9%8C_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="เมทริกซ์ (คณิตศาสตร์) - thailandese" lang="th" hreflang="th" data-title="เมทริกซ์ (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Matris_(matematika)" title="Matris (matematika) - tagalog" lang="tl" hreflang="tl" data-title="Matris (matematika)" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matris_(matematik)" title="Matris (matematik) - turco" lang="tr" hreflang="tr" data-title="Matris (matematik)" 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%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матриця (математика) - ucraino" lang="uk" hreflang="uk" data-title="Матриця (математика)" 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%85%DB%8C%D9%B9%D8%B1%DA%A9%D8%B3_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="میٹرکس (ریاضی) - urdu" lang="ur" hreflang="ur" data-title="میٹرکس (ریاضی)" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matritsa_matematikada" title="Matritsa matematikada - uzbeco" lang="uz" hreflang="uz" data-title="Matritsa matematikada" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ma_tr%E1%BA%ADn_(to%C3%A1n_h%E1%BB%8Dc)" title="Ma trận (toán học) - vietnamita" lang="vi" hreflang="vi" data-title="Ma trận (toán học)" 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-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5" title="矩阵 - wu" lang="wuu" hreflang="wuu" data-title="矩阵" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="voce di qualità"><a href="https://zh.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5" title="矩阵 - cinese" lang="zh" hreflang="zh" data-title="矩阵" 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-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%9F%A9%E9%99%A3" title="矩陣 - cinese classico" lang="lzh" hreflang="lzh" data-title="矩陣" data-language-autonym="文言" data-language-local-name="cinese classico" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/H%C3%A2ng-lia%CC%8Dt" title="Hâng-lia̍t - min nan" lang="nan" hreflang="nan" data-title="Hâng-lia̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%9F%A9%E9%99%A3" title="矩陣 - cantonese" lang="yue" hreflang="yue" data-title="矩陣" 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/Q44337#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul 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.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 nota-disambigua"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/18px-Nota_disambigua.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/27px-Nota_disambigua.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/36px-Nota_disambigua.svg.png 2x" data-file-width="200" data-file-height="200" /></span></span> <span class="hatnote-text"><a href="/wiki/Aiuto:Disambiguazione" title="Aiuto:Disambiguazione">Disambiguazione</a> – Se stai cercando altri significati, vedi <b><a href="/wiki/Matrice_(disambigua)" class="mw-disambig" title="Matrice (disambigua)">Matrice (disambigua)</a></b>.</span></div> </div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Matrix_-_it.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Matrix_-_it.svg/291px-Matrix_-_it.svg.png" decoding="async" width="291" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Matrix_-_it.svg/437px-Matrix_-_it.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Matrix_-_it.svg/582px-Matrix_-_it.svg.png 2x" data-file-width="300" data-file-height="210" /></a><figcaption>Gli elementi di una matrice vengono in genere indicati con una coppia di indici a pedice.</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>, una <b>matrice</b> è una tabella ordinata di elementi. </p><p>Ad esempio: </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{pmatrix}1&0&5\\1&-3&0\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&0&5\\1&-3&0\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d21e16cc2cba1f6696c2d1a89f013d7edd0baed8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.76ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&0&5\\1&-3&0\end{pmatrix}}.}"></span></dd></dl> <p>Le matrici sono ampiamente usate in matematica e in tutte le scienze per la loro capacità di rappresentare in maniera utile e concisa diversi oggetti matematici, come valori che dipendono da due <a href="/wiki/Parametro_(matematica)" title="Parametro (matematica)">parametri</a> o anche <a href="/wiki/Sistema_lineare" class="mw-redirect" title="Sistema lineare">sistemi lineari</a>, cosa, quest'ultima, che le rende uno strumento centrale dell'<a href="/wiki/Analisi_matematica" title="Analisi matematica">analisi matematica</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Storia">Storia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=1" title="Modifica la sezione Storia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=1" title="Edit section's source code: Storia"><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><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/Storia_del_determinante" title="Storia del determinante">Storia del determinante</a></b>.</span></div> </div> <p>Tracce dell'utilizzo di matrici risalgono fino ai primi secoli a.C. Nel corso della storia più volte è capitato che matematici vissuti in epoche e luoghi diversi, durante lo studio di sistemi lineari, abbiano disposto i coefficienti del sistema in forma tabellare, fatto che evidenzia come le matrici siano una struttura particolarmente intuitiva e conveniente per questi scopi.<sup id="cite_ref-mactutor_1-0" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Interessanti reperti sono anche i <a href="/wiki/Quadrato_latino" title="Quadrato latino">quadrati latini</a> e i <a href="/wiki/Quadrato_magico" title="Quadrato magico">quadrati magici</a>. Fu solo a partire dal XVII secolo comunque che l'idea delle matrici fu ripresa e sviluppata, prima con risultati e idee ottenuti in contesti di studio specifici, poi con la loro generalizzazione. Lo sviluppo infine è continuato fino a dare alla teoria delle matrici la forma che oggi conosciamo.<sup id="cite_ref-mactutor_1-1" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>I primi a sfruttare le matrici per agevolare i propri calcoli furono i matematici cinesi, proprio nell'affrontare i sistemi lineari. Nel <i>Jiuzhang Suanshu</i> (<i>Nove capitoli sulle arti matematiche</i>), steso durante la <a href="/wiki/Dinastia_Han" title="Dinastia Han">dinastia Han</a>, l'ottavo capitolo è interamente dedicato allo svolgimento di un problema matematico formulato sotto forma di sistema lineare. L'autore dispone ingegnosamente i coefficienti di ogni equazione parallelamente in senso verticale, in maniera quindi differente dalla notazione odierna, che li vuole disposti orizzontalmente, per righe: una semplice differenza di notazione.<sup id="cite_ref-mactutor_1-2" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-mactutor9_2-0" class="reference"><a href="#cite_note-mactutor9-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Ai numeri così disposti si applicava una serie di operazioni portandoli in una forma tale da rendere evidente quale fosse la soluzione del sistema: è quello che oggi conosciamo come <a href="/wiki/Metodo_di_eliminazione_gaussiana" class="mw-redirect" title="Metodo di eliminazione gaussiana">metodo di eliminazione gaussiana</a>, scoperto in occidente solo agli inizi del XIX secolo con gli studi del matematico tedesco <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>.<sup id="cite_ref-mactutor_1-3" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> All'interno dello <i>Jiuzhang Suanshu</i> comparve anche il concetto di <a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">determinante</a>, inteso come metodo per determinare se un sistema lineare ammette un'unica soluzione.<sup id="cite_ref-mactutor9_2-1" class="reference"><a href="#cite_note-mactutor9-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Un'idea più moderna di determinante fece la sua comparsa nel 1683, a distanza di poco tempo sia in <a href="/wiki/Giappone" title="Giappone">Giappone</a>, con <a href="/wiki/K%C5%8Dwa_Seki" title="Kōwa Seki">Kōwa Seki</a> (<i>Method of solving the dissimulated problems</i>), che in <a href="/wiki/Europa" title="Europa">Europa</a>, con <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a>. Nella prima metà del XVIII secolo, <a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Maclaurin</a> scrisse il <i>Treatise of Algebra</i> (<i>Trattato di algebra</i>)<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>, pubblicato postumo nel <a href="/wiki/1748" title="1748">1748</a>, nella quale mostrava il calcolo dei determinanti per <a href="/wiki/Matrice_quadrata" title="Matrice quadrata">matrici quadrate</a> di ordine 2 e 3. <a href="/wiki/Gabriel_Cramer" title="Gabriel Cramer">Cramer</a> diede il suo contributo nel <a href="/wiki/1750" title="1750">1750</a> presentando l'algoritmo per il calcolo del determinante per matrici quadrate di ordine qualunque, usato nel metodo oggi noto come <a href="/wiki/Regola_di_Cramer" title="Regola di Cramer">regola di Cramer</a> (<i>Introduction à l'analyse des lignes courbes algébriques</i>). Ulteriori sviluppi sul concetto di determinante furono poi apportati da <a href="/wiki/%C3%89tienne_B%C3%A9zout" title="Étienne Bézout">Bézout</a> (<i>Sur le degré des équations résultantes de l'évanouissement des inconnues</i>, 1764), <a href="/wiki/Alexandre-Th%C3%A9ophile_Vandermonde" title="Alexandre-Théophile Vandermonde">Vandermonde</a> (<i>Mémoire sur l'élimination</i>, 1772)<sup id="cite_ref-mactutorv_4-0" class="reference"><a href="#cite_note-mactutorv-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>, <a href="/wiki/Pierre_Simon_Laplace" title="Pierre Simon Laplace">Laplace</a> (1772), <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a> (1773), <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> (1801) che introdusse per la prima volta il termine <i>determinante</i>, <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a> (1812) che usò per la prima volta il determinante nella sua concezione moderna, ottenendo anche importanti risultati sui <a href="/wiki/Minore_(algebra_lineare)" title="Minore (algebra lineare)">minori</a> e le <a href="/wiki/Matrice_trasposta_coniugata" title="Matrice trasposta coniugata">matrici aggiunte</a>, e <a href="/wiki/Carl_Jacobi" title="Carl Jacobi">Jacobi</a>.<sup id="cite_ref-mactutor_1-4" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> All'inizio del <a href="/wiki/XIX_secolo" title="XIX secolo">XIX secolo</a> venne usato per la prima volta in occidente il <a href="/wiki/Metodo_di_eliminazione_gaussiana" class="mw-redirect" title="Metodo di eliminazione gaussiana">metodo di eliminazione gaussiana</a> da parte di Gauss, per lo studio dell'orbita dell'<a href="/wiki/2_Pallas" title="2 Pallas">asteroide Pallas</a> in base alle osservazioni ottenute fra il 1803 ed il 1809.<sup id="cite_ref-mactutor_1-5" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Altri concetti ed idee fondamentali della teoria delle matrici furono poi studiati, sempre in contesti specifici, da Cauchy, <a href="/wiki/Jacques_Charles_Fran%C3%A7ois_Sturm" title="Jacques Charles François Sturm">Sturm</a>, Jacobi, <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Kronecker</a>, <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass</a> e <a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Eisenstein</a>. </p><p>Nel 1848 il matematico e avvocato inglese <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">Sylvester</a> introdusse per la prima volta il termine <i>matrice</i>. Il suo collega avvocato <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a> introdusse nel <a href="/wiki/1853" title="1853">1853</a> l'<a href="/wiki/Matrice_inversa" class="mw-redirect" title="Matrice inversa">inversa</a> di una matrice.<sup id="cite_ref-mactutor_1-6" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>, e nel <a href="/wiki/1858" title="1858">1858</a> fornì la prima definizione astratta di matrice, in <i>Memoir on the theory of matrices</i> (<i>Memorie sulla teoria delle matrici</i>)<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>, mostrando come tutti gli studi precedenti non fossero altro che casi specifici del suo concetto generale. All'interno del testo Cayley forniva inoltre un'algebra delle matrici, definendo le operazioni basilari di somma, moltiplicazione tra matrici, moltiplicazione per scalari e inversa di una matrice.<sup id="cite_ref-mactutor_1-7" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Ancora ignaro di tali opere, nel 1878 <a href="/wiki/Ferdinand_Georg_Frobenius" title="Ferdinand Georg Frobenius">Frobenius</a> pubblicò <i>Ueber lineare Substitutionen und bilineare Formen</i> (<i>Sulle sostituzioni lineari e forme bilineari</i>), nel quale riportava importanti risultati sulle matrici, quale per esempio la definizione di <a href="/wiki/Rango_(algebra_lineare)" title="Rango (algebra lineare)">rango</a><sup id="cite_ref-mactutor_1-8" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. Nel 1888 il geodeta <a href="/wiki/Wilhelm_Jordan" title="Wilhelm Jordan">Jordan</a> nella terza edizione del suo <i>Handbuch der Vermessungskunde</i> (<i>Manuale di geodesia</i>) ampliò il metodo di eliminazione di Gauss in quello che oggi è noto come <a href="/wiki/Metodo_di_eliminazione_di_Gauss-Jordan" class="mw-redirect" title="Metodo di eliminazione di Gauss-Jordan">metodo di eliminazione di Gauss-Jordan</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> Altri contributi importanti furono dati da <a href="/w/index.php?title=Maxime_B%C3%B4cher&action=edit&redlink=1" class="new" title="Maxime Bôcher (la pagina non esiste)">Bôcher</a> nel 1907 con <i>Introduction to higher algebra</i>; altri testi di rilievo furono scritti da <a href="/w/index.php?title=Herbert_Westren_Turnbull&action=edit&redlink=1" class="new" title="Herbert Westren Turnbull (la pagina non esiste)">Turnbull</a> ed <a href="/wiki/Alexander_Craig_Aitken" class="mw-redirect" title="Alexander Craig Aitken">Aitken</a> negli <a href="/wiki/Anni_1930" title="Anni 1930">anni trenta</a> (<i>The Theory of Canonical Matrices</i> e <i>Determinants and Matrices</i>) e da <a href="/w/index.php?title=Leon_Mirsky&action=edit&redlink=1" class="new" title="Leon Mirsky (la pagina non esiste)">Mirsky</a> nel 1955 (<i>An introduction to linear algebra</i>).<sup id="cite_ref-mactutor_1-9" class="reference"><a href="#cite_note-mactutor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>A partire dalla seconda metà del <a href="/wiki/XX_secolo" title="XX secolo">XX secolo</a> l'avvento dei <a href="/wiki/Computer" title="Computer">computer</a> ha dato un'impressionante accelerazione alla diffusione delle matrici e dei metodi matriciali. Grazie ai computer infatti è stato possibile applicare in maniera efficiente <a href="/wiki/Metodo_iterativo" title="Metodo iterativo">metodi iterativi</a> precedentemente ritenuti troppo onerosi, portando di conseguenza allo sviluppo di nuove tecniche per la risoluzione di importanti problemi dell'algebra lineare, quali il calcolo degli <a href="/wiki/Autovettore_e_autovalore" title="Autovettore e autovalore">autovettori e autovalori</a>, il calcolo dell'<a href="/wiki/Matrice_inversa" class="mw-redirect" title="Matrice inversa">inversa</a> di una matrice e la risoluzione di sistemi lineari.<sup id="cite_ref-bro89-1_7-0" class="reference"><a href="#cite_note-bro89-1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Ciò a sua volta ha permesso l'introduzione delle matrici in altre discipline applicate, come per esempio la <a href="/w/index.php?title=Matematica_economica&action=edit&redlink=1" class="new" title="Matematica economica (la pagina non esiste)">matematica economica</a> e la <a href="/wiki/Teoria_della_probabilit%C3%A0" title="Teoria della probabilità">probabilità</a>, che grazie ad esse hanno potuto rappresentare concetti complessi in maniera più semplice. Altri campi relativamente più recenti, invece, come per esempio la <a href="/wiki/Ricerca_operativa" title="Ricerca operativa">ricerca operativa</a>, hanno basato ampiamente la propria disciplina sull'utilizzo delle matrici.<sup id="cite_ref-bro89-1_7-1" class="reference"><a href="#cite_note-bro89-1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definizioni_e_notazioni">Definizioni e notazioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=2" title="Modifica la sezione Definizioni e notazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=2" title="Edit section's source code: Definizioni e notazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una <b>matrice</b> è una tabella rettangolare di numeri. Da un punto di vista formale, può essere definita come una 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 A\colon \{1,\ldots ,m\}\times \{1,\ldots ,n\}\to K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo fence="false" stretchy="false">}</mo> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">→</mo> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\colon \{1,\ldots ,m\}\times \{1,\ldots ,n\}\to K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c727d82f763c5e6257042aa819272ba7b8b3fa9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.711ex; height:2.843ex;" alt="{\displaystyle A\colon \{1,\ldots ,m\}\times \{1,\ldots ,n\}\to K,}"></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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></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 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> sono <a href="/wiki/Numero_intero" title="Numero intero">interi</a> positivi fissati 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}"> <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 un qualunque <a href="/wiki/Insieme" title="Insieme">insieme</a> fissato, come ad esempio quello dei <a href="/wiki/Numeri_reali" class="mw-redirect" title="Numeri reali">numeri reali</a>. Le righe orizzontali di una matrice sono chiamate <i>righe</i>, mentre quelle verticali <i>colonne</i>. Ad esempio, la matrice mostrata sopra ha due righe e tre colonne. Una 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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> generica è descritta come in figura sopra oppure anche nel modo seguente (che viene considerata più proficua come notazione per il fatto di non dover differenziare nelle operazioni l'elemento dalla matrice stessa): </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{pmatrix}\;[A]_{1,1}&[A]_{1,2}&\cdots &[A]_{1,n}\\\;[A]_{2,1}&[A]_{2,2}&\cdots &[A]_{2,n}\\\vdots &\vdots &\ddots &\vdots \\\;[A]_{m,1}&[A]_{m,2}&\cdots &[A]_{m,n}\end{pmatrix}}}"> <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" /> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}\;[A]_{1,1}&[A]_{1,2}&\cdots &[A]_{1,n}\\\;[A]_{2,1}&[A]_{2,2}&\cdots &[A]_{2,n}\\\vdots &\vdots &\ddots &\vdots \\\;[A]_{m,1}&[A]_{m,2}&\cdots &[A]_{m,n}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25fe31f3586dc651fc298ce42b6c527a3304a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:38.389ex; height:14.843ex;" alt="{\displaystyle A={\begin{pmatrix}\;[A]_{1,1}&[A]_{1,2}&\cdots &[A]_{1,n}\\\;[A]_{2,1}&[A]_{2,2}&\cdots &[A]_{2,n}\\\vdots &\vdots &\ddots &\vdots \\\;[A]_{m,1}&[A]_{m,2}&\cdots &[A]_{m,n}\end{pmatrix}}}"></span></dd></dl> <p>indicando 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 [A]_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [A]_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b112d2151a8c1e6f631983e20f07d52e80f6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.971ex; height:3.009ex;" alt="{\displaystyle [A]_{i,j}}"></span> l'elemento posizionato alla riga <span class="mwe-math-element"><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/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-esima e nella colonna <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>-esima. </p><p>La riga <span class="mwe-math-element"><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/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-esima viene indicata 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 Row_{i}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>o</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Row_{i}(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac45e0ddde5877344ec976973bd2e4488f8d4846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.908ex; height:2.843ex;" alt="{\displaystyle Row_{i}(A)}"></span>, oppure più ambiguamente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"></span>, mentre colonna <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>-esima 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 Col_{j}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>o</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Col_{j}(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf22cac145347ac358eafe5f959e064e1ff3dbe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.049ex; height:3.009ex;" alt="{\displaystyle Col_{j}(A)}"></span>, oppure più ambiguamente <span class="mwe-math-element"><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^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/447068b028400a3e454b3b8cd8dd4ba444ab25e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.653ex; height:2.676ex;" alt="{\displaystyle A^{j}}"></span>. </p><p>Gli elementi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [A]_{i,i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [A]_{i,i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f9947cb832d93bec91a5492bb8846b870bc5dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.861ex; height:3.009ex;" alt="{\displaystyle [A]_{i,i}}"></span> costituiscono la <a href="/wiki/Diagonale_principale" title="Diagonale principale">diagonale principale</a> della matrice. </p><p>I <a href="/wiki/Vettore_(matematica)" title="Vettore (matematica)">vettori</a> possono essere considerati matrici aventi una sola riga o una sola colonna. Una matrice con una sola riga, 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 1\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bce5f6a6d0d32834484048c16f3b39f9c23d076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle 1\times n}"></span>, è detta <i>matrice riga</i> o <i>vettore riga</i>, mentre una matrice con una sola colonna, 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 m\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7ea91d81567531f5ef6d3b669be211ff953e6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.043ex; height:2.176ex;" alt="{\displaystyle m\times 1}"></span>, è detta <i>matrice colonna</i> o <i>vettore colonna</i>. </p><p>Di seguito sono mostrati in ordine una 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 4\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49de1a2b7cbca0fccbb7e119e68fd225383da5a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 4\times 3}"></span>, una matrice colonna ed una matrice riga; </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{pmatrix}1&2&3\\1&-2&0\\4,5&0&2\\6&1&5\end{pmatrix}},\quad {\begin{pmatrix}7\\0\\\pi \end{pmatrix}},\quad {\begin{pmatrix}3&{\frac {7}{2}}&-9\end{pmatrix}}.}"> <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>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mn>9</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&2&3\\1&-2&0\\4,5&0&2\\6&1&5\end{pmatrix}},\quad {\begin{pmatrix}7\\0\\\pi \end{pmatrix}},\quad {\begin{pmatrix}3&{\frac {7}{2}}&-9\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fedc7b72cefa4fdf5ee87624e3299f7842874f94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:44.429ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1&2&3\\1&-2&0\\4,5&0&2\\6&1&5\end{pmatrix}},\quad {\begin{pmatrix}7\\0\\\pi \end{pmatrix}},\quad {\begin{pmatrix}3&{\frac {7}{2}}&-9\end{pmatrix}}.}"></span></dd></dl> <p>Come mostrato negli esempi, i valori presenti nella matrice possono essere di vario tipo: <a href="/wiki/Numero_intero" title="Numero intero">interi</a>, <a href="/wiki/Numero_reale" title="Numero reale">reali</a> o anche <a href="/wiki/Numero_complesso" title="Numero complesso">complessi</a>. In molti casi si suppone che i valori siano elementi di 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> fissato. </p> <div class="mw-heading mw-heading2"><h2 id="Algebra_delle_matrici">Algebra delle matrici</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=3" title="Modifica la sezione Algebra delle matrici" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=3" title="Edit section's source code: Algebra delle matrici"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sulle matrici si possono definire numerose operazioni che spesso dipendono anche dall'insieme in cui sono scelti i valori delle matrici. Nel resto del paragrafo supponiamo che le matrici abbiano tutte valori in uno stesso campo <span class="mwe-math-element"><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> fissato. </p> <div class="mw-heading mw-heading3"><h3 id="Somma">Somma</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=4" title="Modifica la sezione Somma" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=4" title="Edit section's source code: Somma"><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/Somma_fra_matrici" class="mw-redirect" title="Somma fra matrici">Somma fra matrici</a></b>.</span></div> </div> <p>Due matrici <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, entrambe di tipo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span>, possono essere sommate. La loro somma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A+B}"></span> è definita come 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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> i cui elementi sono ottenuti sommando i corrispettivi elementi 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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. Formalmente: </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+B]_{i,j}:=[A]_{i,j}+[B]_{i,j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>:=</mo> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <mi>B</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [A+B]_{i,j}:=[A]_{i,j}+[B]_{i,j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/618ebc3a577e9f52039846a9003cb5c8d63e16e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.772ex; height:3.009ex;" alt="{\displaystyle [A+B]_{i,j}:=[A]_{i,j}+[B]_{i,j}.}"></span></dd></dl> <p>Per esempio: </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{pmatrix}1&3&2\\1&0&0\\1&2&2\end{pmatrix}}+{\begin{pmatrix}0&0&5\\7&5&0\\2&1&1\end{pmatrix}}={\begin{pmatrix}1+0&3+0&2+5\\1+7&0+5&0+0\\1+2&2+1&2+1\end{pmatrix}}={\begin{pmatrix}1&3&7\\8&5&0\\3&3&3\end{pmatrix}}.}"> <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>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>2</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>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <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>1</mn> <mo>+</mo> <mn>0</mn> </mtd> <mtd> <mn>3</mn> <mo>+</mo> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mo>+</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <mn>7</mn> </mtd> <mtd> <mn>0</mn> <mo>+</mo> <mn>5</mn> </mtd> <mtd> <mn>0</mn> <mo>+</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <mn>2</mn> </mtd> <mtd> <mn>2</mn> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mn>2</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> <mtd> <mn>3</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&3&2\\1&0&0\\1&2&2\end{pmatrix}}+{\begin{pmatrix}0&0&5\\7&5&0\\2&1&1\end{pmatrix}}={\begin{pmatrix}1+0&3+0&2+5\\1+7&0+5&0+0\\1+2&2+1&2+1\end{pmatrix}}={\begin{pmatrix}1&3&7\\8&5&0\\3&3&3\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66f7d489826f960e303a6380ed430ab587ce6f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:73.496ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}1&3&2\\1&0&0\\1&2&2\end{pmatrix}}+{\begin{pmatrix}0&0&5\\7&5&0\\2&1&1\end{pmatrix}}={\begin{pmatrix}1+0&3+0&2+5\\1+7&0+5&0+0\\1+2&2+1&2+1\end{pmatrix}}={\begin{pmatrix}1&3&7\\8&5&0\\3&3&3\end{pmatrix}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Moltiplicazione_per_uno_scalare">Moltiplicazione per uno scalare</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=5" title="Modifica la sezione Moltiplicazione per uno scalare" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=5" title="Edit section's source code: Moltiplicazione per uno scalare"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La moltiplicazione per uno scalare è un'operazione che, data una 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 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> e un numero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> (detto <i>scalare</i>), costruisce una nuova 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 c\cdot A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>⋅</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\cdot A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea5b6be3d6d08fd68b4715580ca258b7b63fc8ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.429ex; height:2.176ex;" alt="{\displaystyle c\cdot A}"></span>, il cui elemento è ottenuto moltiplicando l'elemento corrispondente 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> 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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>; gli elementi della matrice e lo scalare in questione devono appartenere allo stesso <a href="/wiki/Campo_(matematica)" title="Campo (matematica)">campo</a>. Formalmente: </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 [cA]_{ij}:=c[A]_{i,j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>c</mi> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>:=</mo> <mi>c</mi> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [cA]_{ij}:=c[A]_{i,j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfba89f5d8c48e3ce8b3cc3433d23e365dc3e8a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.891ex; height:3.009ex;" alt="{\displaystyle [cA]_{ij}:=c[A]_{i,j}.}"></span></dd></dl> <p>Per esempio: </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 2{\begin{pmatrix}1&8&-3\\4&-2&5\end{pmatrix}}={\begin{pmatrix}2\cdot 1&2\cdot 8&2\cdot (-3)\\2\cdot 4&2\cdot (-2)&2\cdot 5\end{pmatrix}}={\begin{pmatrix}2&16&-6\\8&-4&10\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</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>2</mn> <mo>⋅</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>8</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>5</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>2</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mn>10</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\begin{pmatrix}1&8&-3\\4&-2&5\end{pmatrix}}={\begin{pmatrix}2\cdot 1&2\cdot 8&2\cdot (-3)\\2\cdot 4&2\cdot (-2)&2\cdot 5\end{pmatrix}}={\begin{pmatrix}2&16&-6\\8&-4&10\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb176b7334a2c05635b3fb20744dfc69e5b58910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.914ex; height:6.176ex;" alt="{\displaystyle 2{\begin{pmatrix}1&8&-3\\4&-2&5\end{pmatrix}}={\begin{pmatrix}2\cdot 1&2\cdot 8&2\cdot (-3)\\2\cdot 4&2\cdot (-2)&2\cdot 5\end{pmatrix}}={\begin{pmatrix}2&16&-6\\8&-4&10\end{pmatrix}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Prodotto">Prodotto</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=6" title="Modifica la sezione Prodotto" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=6" title="Edit section's source code: Prodotto"><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/Moltiplicazione_di_matrici" title="Moltiplicazione di matrici">Moltiplicazione di matrici</a></b>.</span></div> </div> <p>La moltiplicazione tra due matrici <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> è un'operazione più complessa delle precedenti. A differenza della somma, non avviene moltiplicando semplicemente gli elementi aventi lo stesso posto. La definizione di moltiplicazione che segue è motivata dal fatto che una matrice modellizza un'<a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">applicazione lineare</a>, e il prodotto di matrici corrisponde alla <a href="/wiki/Composizione_di_funzioni" title="Composizione di funzioni">composizione</a> di applicazioni lineari. </p><p>La moltiplicazione è quindi definita soltanto se le matrici <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> sono rispettivamente di tipo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e459f087ef822dc6fba54b953c60de61be69c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.05ex; height:2.009ex;" alt="{\displaystyle m\times p}"></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 p\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2bd3ebfae88cd66c5c2eb301e54467793956fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.494ex; height:2.009ex;" alt="{\displaystyle p\times n}"></span>: in altre parole, il numero <span class="mwe-math-element"><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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> di colonne 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> deve coincidere con il numero <span class="mwe-math-element"><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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. Il risultato è una 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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> di tipo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span>. </p><p>Ad esempio, siano <span class="mwe-math-element"><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> 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> due matrici rispettivamente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fda443e7a6e78fa880a6dccbf8bdf43a10d9988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 4}"></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 4\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ef3ad1c66c8e543248197c5ecc2676627053c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 4\times 2}"></span>: tra queste si può effettuare la moltiplicazione <span class="mwe-math-element"><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\times B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f31ae45b0098f06b5d22c38d317eb097a88fa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.348ex; height:2.176ex;" alt="{\displaystyle A\times B}"></span> ed ottenere una 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 3\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac4e0cc6e76e3127e7dbcc26735b6925690997eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 2}"></span>. Le stesse matrici, però, non possono essere moltiplicate nel modo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\times A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>×</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\times A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1f7f117a7b4c40424838eb9f4717e683c46b2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.348ex; height:2.176ex;" alt="{\displaystyle B\times A}"></span>, poiché le colonne 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> non sono tante quante le 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>. </p><p>Il prodotto 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> 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> righe 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 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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> colonne 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> righe 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 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> colonne è 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 C=AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636d3b809b596fe5cd46c13aff3deb285f18cba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.372ex; height:2.176ex;" alt="{\displaystyle C=AB}"></span> 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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span>, il cui elemento di posizione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}"></span> è dato dalla somma: </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 [C]_{i,j}=Row_{i}(A)\times Col_{j}(B)=[A]_{i,1}[B]_{1,j}+[A]_{i,2}[B]_{2,j}+\cdots +[A]_{i,p}[B]_{p,j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>R</mi> <mi>o</mi> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>C</mi> <mi>o</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>B</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>B</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>B</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [C]_{i,j}=Row_{i}(A)\times Col_{j}(B)=[A]_{i,1}[B]_{1,j}+[A]_{i,2}[B]_{2,j}+\cdots +[A]_{i,p}[B]_{p,j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9f7acbf171132aeb3d64ec162e049f9e888920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:73.999ex; height:3.009ex;" alt="{\displaystyle [C]_{i,j}=Row_{i}(A)\times Col_{j}(B)=[A]_{i,1}[B]_{1,j}+[A]_{i,2}[B]_{2,j}+\cdots +[A]_{i,p}[B]_{p,j}.}"></span></dd></dl> <p>Quest'ultimo viene detto <i>prodotto riga per colonna</i>. Ad esempio: </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{pmatrix}1&1&2\\0&1&-3\end{pmatrix}},\qquad B={\begin{pmatrix}1&1&1\\2&5&1\\0&-2&1\end{pmatrix}}}"> <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>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}1&1&2\\0&1&-3\end{pmatrix}},\qquad B={\begin{pmatrix}1&1&1\\2&5&1\\0&-2&1\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66283d2475774549955cd64b108dcb9a3a026536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:44.255ex; height:9.176ex;" alt="{\displaystyle A={\begin{pmatrix}1&1&2\\0&1&-3\end{pmatrix}},\qquad B={\begin{pmatrix}1&1&1\\2&5&1\\0&-2&1\end{pmatrix}}}"></span></dd> <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 C=AB={\begin{pmatrix}1&1&2\\0&1&-3\end{pmatrix}}{\begin{pmatrix}1&1&1\\2&5&1\\0&-2&1\end{pmatrix}}={\begin{pmatrix}\,\![C]_{1,1}&[C]_{1,2}&[C]_{1,3}\\\,\![C]_{2,1}&[C]_{2,2}&[C]_{2,3}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <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>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=AB={\begin{pmatrix}1&1&2\\0&1&-3\end{pmatrix}}{\begin{pmatrix}1&1&1\\2&5&1\\0&-2&1\end{pmatrix}}={\begin{pmatrix}\,\![C]_{1,1}&[C]_{1,2}&[C]_{1,3}\\\,\![C]_{2,1}&[C]_{2,2}&[C]_{2,3}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edd5f6ca7079fbf1636dde3444bf3bf2185bcbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:69.087ex; height:9.176ex;" alt="{\displaystyle C=AB={\begin{pmatrix}1&1&2\\0&1&-3\end{pmatrix}}{\begin{pmatrix}1&1&1\\2&5&1\\0&-2&1\end{pmatrix}}={\begin{pmatrix}\,\![C]_{1,1}&[C]_{1,2}&[C]_{1,3}\\\,\![C]_{2,1}&[C]_{2,2}&[C]_{2,3}\end{pmatrix}}.}"></span></dd></dl> <p>Si osserva che moltiplicando una 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 2\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b74d5a424cfb56b99e1060910dbfed284314da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 3}"></span> per una <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc0d4d6106875f8006be1d898512ca5843bad8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}"></span> si ottiene una 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 2\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b74d5a424cfb56b99e1060910dbfed284314da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 3}"></span>. </p><p>Prima riga: </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 \![C]_{1,1}=(1\cdot 1)+(1\cdot 2)+(2\cdot 0)=3;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="negativethinmathspace" /> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \![C]_{1,1}=(1\cdot 1)+(1\cdot 2)+(2\cdot 0)=3;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/701c5b4797380b99274785a83a64c1a56d39e422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.387ex; width:36.521ex; height:3.009ex;" alt="{\displaystyle \![C]_{1,1}=(1\cdot 1)+(1\cdot 2)+(2\cdot 0)=3;}"></span></dd> <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 [C]_{1,2}=(1\cdot 1)+(1\cdot 5)+(2\cdot (-2))=2;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [C]_{1,2}=(1\cdot 1)+(1\cdot 5)+(2\cdot (-2))=2;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c62d8a09a4ba3f308a1ce71800ffe8be1caaa610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.138ex; height:3.009ex;" alt="{\displaystyle [C]_{1,2}=(1\cdot 1)+(1\cdot 5)+(2\cdot (-2))=2;}"></span></dd> <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 [C]_{1,3}=(1\cdot 1)+(1\cdot 1)+(2\cdot 1)=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [C]_{1,3}=(1\cdot 1)+(1\cdot 1)+(2\cdot 1)=4.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07731fa772db96ed35c3f16bf6a933d41e4c5e68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.521ex; height:3.009ex;" alt="{\displaystyle [C]_{1,3}=(1\cdot 1)+(1\cdot 1)+(2\cdot 1)=4.}"></span></dd></dl> <p>Seconda riga: </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 [C]_{2,1}=(0\cdot 1)+(1\cdot 2)+(-3\cdot 0)=2;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo>⋅</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [C]_{2,1}=(0\cdot 1)+(1\cdot 2)+(-3\cdot 0)=2;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a80c2ffcdde0ab55174e5fb5e434789270bb7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.329ex; height:3.009ex;" alt="{\displaystyle [C]_{2,1}=(0\cdot 1)+(1\cdot 2)+(-3\cdot 0)=2;}"></span></dd> <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 [C]_{2,2}=(0\cdot 1)+(1\cdot 5)+(-3\cdot (-2))=11;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>11</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [C]_{2,2}=(0\cdot 1)+(1\cdot 5)+(-3\cdot (-2))=11;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f7c7afbb3d524d841be16f51a3f353356fd85b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.109ex; height:3.009ex;" alt="{\displaystyle [C]_{2,2}=(0\cdot 1)+(1\cdot 5)+(-3\cdot (-2))=11;}"></span></dd> <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 [C]_{2,3}=(0\cdot 1)+(1\cdot 1)+(-3\cdot 1)=-2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [C]_{2,3}=(0\cdot 1)+(1\cdot 1)+(-3\cdot 1)=-2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e3a3024967ae397bf6bada711d0736d1d627d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.137ex; height:3.009ex;" alt="{\displaystyle [C]_{2,3}=(0\cdot 1)+(1\cdot 1)+(-3\cdot 1)=-2.}"></span></dd></dl> <p>Da cui: </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 C={\begin{pmatrix}\,\![C]_{1,1}&[C]_{1,2}&[C]_{1,3}\\\,\![C]_{2,1}&[C]_{2,2}&[C]_{2,3}\end{pmatrix}}={\begin{pmatrix}3&2&4\\2&11&-2\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">[</mo> <mi>C</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>11</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\begin{pmatrix}\,\![C]_{1,1}&[C]_{1,2}&[C]_{1,3}\\\,\![C]_{2,1}&[C]_{2,2}&[C]_{2,3}\end{pmatrix}}={\begin{pmatrix}3&2&4\\2&11&-2\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d3db85ae6e75acccc8c7a77990c9dcaeeae652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.885ex; height:6.509ex;" alt="{\displaystyle C={\begin{pmatrix}\,\![C]_{1,1}&[C]_{1,2}&[C]_{1,3}\\\,\![C]_{2,1}&[C]_{2,2}&[C]_{2,3}\end{pmatrix}}={\begin{pmatrix}3&2&4\\2&11&-2\end{pmatrix}}.}"></span></dd></dl> <p>A differenza dell'usuale moltiplicazione fra numeri, questa non è un'operazione <a href="/wiki/Propriet%C3%A0_commutativa" class="mw-redirect" title="Proprietà commutativa">commutativa</a>, 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 AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"></span> è in generale diverso 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 BA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8efb97e621ab9b49f8498a49704690bdeb2698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle BA}"></span>, quando si possono effettuare entrambi questi prodotti. </p><p>Un caso particolare, ampiamente usato in <a href="/wiki/Algebra_lineare" title="Algebra lineare">algebra lineare</a> per rappresentare le <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazioni lineari</a> (come <a href="/wiki/Rotazione_(matematica)" title="Rotazione (matematica)">rotazioni</a> e <a href="/wiki/Riflessione_(geometria)" title="Riflessione (geometria)">riflessioni</a>) è il prodotto tra una 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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> ed un vettore colonna <span class="mwe-math-element"><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\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24148f103e1cccb60addeeb0a64cb1c3d5622e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle n\times 1}"></span>, che viene chiamato anche <i><a href="/wiki/Moltiplicazione_di_matrici#Prodotto_di_una_matrice_per_un_vettore" title="Moltiplicazione di matrici">prodotto matrice-vettore</a></i>. </p> <div class="mw-heading mw-heading3"><h3 id="Proprietà"><span id="Propriet.C3.A0"></span>Proprietà</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=7" title="Modifica la sezione Proprietà" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=7" title="Edit section's source code: Proprietà"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le operazioni di somma e prodotto di matrici soddisfano tutte le proprietà usuali della somma e del prodotto di numeri, ad eccezione, nel caso del prodotto di matrici, della proprietà commutativa. </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 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> la <a href="/wiki/Matrice_nulla" title="Matrice nulla">matrice nulla</a>, fatta di soli zeri (e della stessa taglia 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>). Sia inoltre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -A=(-1)A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -A=(-1)A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22289c34aa537123c7914c2eff94d9ea19bb0400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.173ex; height:2.843ex;" alt="{\displaystyle -A=(-1)A}"></span> la matrice ottenuta moltiplicando <span class="mwe-math-element"><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> per lo 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 -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>. Valgono le relazioni seguenti, per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce2acf22b93dfbd22373336bd9c22dbd98a49d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.509ex;" alt="{\displaystyle A,B,C}"></span> matrici <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> e, per ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> numeri reali. </p> <div class="mw-heading mw-heading4"><h4 id="Proprietà_della_somma_e_del_prodotto_per_uno_scalare"><span id="Propriet.C3.A0_della_somma_e_del_prodotto_per_uno_scalare"></span>Proprietà della somma e del prodotto per uno scalare</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=8" title="Modifica la sezione Proprietà della somma e del prodotto per uno scalare" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=8" title="Edit section's source code: Proprietà della somma e del prodotto per uno scalare"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 A+0=0+A=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mi>A</mi> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+0=0+A=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19222cbea5be011904589719db31ec2dc0246873" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.432ex; height:2.343ex;" alt="{\displaystyle A+0=0+A=A}"></span> (la matrice nulla è l'<a href="/wiki/Elemento_neutro" title="Elemento neutro">elemento neutro</a> della somma);</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 A+(-A)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+(-A)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5d2f29fb29fae380a5ae585ee516f43d3a2f0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.205ex; height:2.843ex;" alt="{\displaystyle A+(-A)=0}"></span> (esistenza di un <a href="/wiki/Elemento_opposto" class="mw-redirect" title="Elemento opposto">opposto</a> per la somma);</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 (A+B)+C=A+(B+C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A+B)+C=A+(B+C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2990c2cd9b82bd081dd880bc9ccb0f3f6567d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.626ex; height:2.843ex;" alt="{\displaystyle (A+B)+C=A+(B+C)}"></span> (<a href="/wiki/Propriet%C3%A0_associativa" class="mw-redirect" title="Proprietà associativa">proprietà associativa</a> della somma);</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 A+B=B+A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>=</mo> <mi>B</mi> <mo>+</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B=B+A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd88bf3e865bc9893426ba275d3890457222142c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.793ex; height:2.343ex;" alt="{\displaystyle A+B=B+A}"></span> (proprietà commutativa della somma);</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 1A=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mi>A</mi> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1A=A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96dcd6b769b1bb0564211664d13cd0e92c475653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.747ex; height:2.176ex;" alt="{\displaystyle 1A=A}"></span> (1 è l'<a href="/wiki/Elemento_neutro" title="Elemento neutro">elemento neutro</a> del prodotto per uno scalare);</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 (ab)A=a(bA)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mi>A</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ab)A=a(bA)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac276e70932cbd1379a555a414d722ed3f01b87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.658ex; height:2.843ex;" alt="{\displaystyle (ab)A=a(bA)}"></span> (proprietà associativa del prodotto per uno scalare);</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 a(A+B)=aA+aB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>A</mi> <mo>+</mo> <mi>a</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(A+B)=aA+aB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45f9157841a04c1b2097cfff57c83c5a0e66240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.292ex; height:2.843ex;" alt="{\displaystyle a(A+B)=aA+aB}"></span> (proprietà distributiva del prodotto per uno scalare rispetto alla somma).</li></ul> <p>Le prime 4 proprietà affermano che le matrici <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> formano un <a href="/wiki/Gruppo_abeliano" title="Gruppo abeliano">gruppo abeliano</a> rispetto all'operazione di somma. Come mostrato sopra, il prodotto non è commutativo in generale. </p> <div class="mw-heading mw-heading4"><h4 id="Proprietà_del_prodotto_fra_matrici"><span id="Propriet.C3.A0_del_prodotto_fra_matrici"></span>Proprietà del prodotto fra matrici</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=9" title="Modifica la sezione Proprietà del prodotto fra matrici" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=9" title="Edit section's source code: Proprietà del prodotto fra matrici"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 (AB)C=A(BC)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mi>C</mi> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (AB)C=A(BC)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e041d54540f5581ab13f2ef4ca2b757033bdc37e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.264ex; height:2.843ex;" alt="{\displaystyle (AB)C=A(BC)}"></span> (proprietà associativa del prodotto);</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 (A+B)C=AC+BC;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mi>C</mi> <mo>=</mo> <mi>A</mi> <mi>C</mi> <mo>+</mo> <mi>B</mi> <mi>C</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A+B)C=AC+BC;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280e7dd04fce92ca5409786728be7038cd3eba16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.549ex; height:2.843ex;" alt="{\displaystyle (A+B)C=AC+BC;}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(A+B)=CA+CB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mi>A</mi> <mo>+</mo> <mi>C</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(A+B)=CA+CB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b308397e88e43d06820dd357e5ff042bbed176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.902ex; height:2.843ex;" alt="{\displaystyle C(A+B)=CA+CB}"></span> (<a href="/wiki/Propriet%C3%A0_distributiva" class="mw-redirect" title="Proprietà distributiva">proprietà distributiva</a> del prodotto rispetto alla somma).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Altre_operazioni">Altre operazioni</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=10" title="Modifica la sezione Altre operazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=10" title="Edit section's source code: Altre operazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sulle matrici sono definite numerose altre operazioni. Tra queste: </p> <ul><li><a href="/wiki/Matrice_trasposta" title="Matrice trasposta">trasposizione</a> di una matrice;</li> <li><a href="/wiki/Somma_fra_matrici#Somma_diretta" class="mw-redirect" title="Somma fra matrici">somma diretta</a>;</li> <li><a href="/wiki/Prodotto_di_Kronecker" title="Prodotto di Kronecker">prodotto diretto</a> (o di <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Kronecker</a>);</li> <li><a href="/wiki/Matrice_esponenziale" title="Matrice esponenziale">matrice esponenziale</a>;</li> <li>inversione di una <a href="/wiki/Matrice_invertibile" title="Matrice invertibile">matrice invertibile</a>;</li> <li>diagonalizzazione di una <a href="/wiki/Matrice_diagonalizzabile" class="mw-redirect" title="Matrice diagonalizzabile">matrice diagonalizzabile</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Applicazioni_lineari">Applicazioni lineari</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=11" title="Modifica la sezione Applicazioni lineari" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=11" title="Edit section's source code: Applicazioni lineari"><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/Matrice_di_trasformazione" title="Matrice di trasformazione">Matrice di trasformazione</a></b> e <b><a href="/wiki/Matrice_di_rotazione" title="Matrice di rotazione">Matrice di rotazione</a></b>.</span></div> </div> <p>Le matrici permettono di rappresentare le <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazioni lineari</a> fra <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazi vettoriali</a>. Ad ogni operatore 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\colon V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\colon V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9307c45fa2557b519d013c5b4229c9bd53a6c3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.507ex; height:2.176ex;" alt="{\displaystyle T\colon V\to W}"></span> da 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> di <a href="/wiki/Dimensione_(spazio_vettoriale)" title="Dimensione (spazio vettoriale)">dimensione</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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> a 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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> 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/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> si associa, per ogni possibile scelta di una coppia di <a href="/wiki/Base_(algebra_lineare)" title="Base (algebra lineare)">basi</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_{1},v_{2},\ldots ,v_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{v_{1},v_{2},\ldots ,v_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3035e8667ba04beea70ab36335780ed766bef5d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.703ex; height:2.843ex;" alt="{\displaystyle \{v_{1},v_{2},\ldots ,v_{m}\}}"></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 \{w_{1},w_{2},\ldots ,w_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{w_{1},w_{2},\ldots ,w_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1506eddd0cafe2090f183fbe5fde77620d533563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.856ex; height:2.843ex;" alt="{\displaystyle \{w_{1},w_{2},\ldots ,w_{n}\}}"></span>, 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 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> 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 T(v_{i})=\sum _{j=1}^{n}[A]_{i,j}w_{j},\quad \forall i\in \{1,2,\ldots ,m\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <mi>A</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(v_{i})=\sum _{j=1}^{n}[A]_{i,j}w_{j},\quad \forall i\in \{1,2,\ldots ,m\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7514b8b6ab7b6fa7e6675a00c66beeae11dc1020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:41.213ex; height:7.176ex;" alt="{\displaystyle T(v_{i})=\sum _{j=1}^{n}[A]_{i,j}w_{j},\quad \forall i\in \{1,2,\ldots ,m\}.}"></span></dd></dl> <p>Questa matrice rappresenta l'applicazione <span class="mwe-math-element"><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> nelle basi scelte. Molte operazioni fra matrici si traducono in operazioni fra applicazioni lineari: </p> <ul><li>l'immagine <span class="mwe-math-element"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d0491c822c9afd420c0a440dcf4a4cf43ebdf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.573ex; height:2.843ex;" alt="{\displaystyle T(v)}"></span> di un vettore corrisponde alla moltiplicazione matrice-vettore;</li> <li>la somma di applicazioni (quando possibile) corrisponde alla somma fra matrici;</li> <li>la <a href="/wiki/Funzione_composta" class="mw-redirect" title="Funzione composta">composizione</a> di applicazioni lineari (quando possibile) corrisponde al prodotto fra matrici.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Sistemi_lineari">Sistemi lineari</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=12" title="Modifica la sezione Sistemi lineari" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=12" title="Edit section's source code: Sistemi lineari"><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/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari">Sistema di equazioni lineari</a></b>.</span></div> </div> <p>Le matrici sono utili soprattutto a rappresentare sistemi di equazioni lineari. Il sistema: </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\{{\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}=b_{m}\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}=b_{m}\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0de83c5e7eda54bc2802800f3c94f2cf2c93eebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:38.494ex; height:13.843ex;" alt="{\displaystyle \left\{{\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}=b_{m}\end{matrix}}\right.}"></span></dd></dl> <p>può essere rappresentato con il suo equivalente matriciale, tramite il <a href="/wiki/Moltiplicazione_di_matrici#Prodotto_di_una_matrice_per_un_vettore" title="Moltiplicazione di matrici">prodotto matrice-vettore</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 {\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{pmatrix}}.}"> <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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92d28173e09623417be6d4276e3d1cd433c3900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:44.712ex; height:14.176ex;" alt="{\displaystyle {\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{pmatrix}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Matrici_quadrate">Matrici quadrate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=13" title="Modifica la sezione Matrici quadrate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=13" title="Edit section's source code: Matrici quadrate"><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/Matrice_quadrata" title="Matrice quadrata">Matrice quadrata</a></b>.</span></div> </div> <p>Fra le matrici, occupano un posto di rilievo le matrici quadrate, cioè le matrici <span class="mwe-math-element"><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\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span>, che hanno lo stesso numero <span class="mwe-math-element"><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 righe e di colonne. Una matrice quadrata ha una <a href="/wiki/Diagonale_principale" title="Diagonale principale">diagonale principale</a>, quella formata da tutti gli elementi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d0b851d01900d6473e1aa9bdaddcc7c764cada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.054ex; height:2.343ex;" alt="{\displaystyle a_{i,i}}"></span> con indici uguali. La somma di questi elementi è chiamata <a href="/wiki/Traccia_(matrice)" title="Traccia (matrice)">traccia</a>. L'operazione di <a href="/wiki/Trasposta_di_una_matrice" class="mw-redirect" title="Trasposta di una matrice">trasposizione</a> trasforma 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> nella 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 A^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mathrm {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0966b7ada93c12ff7d306496f224f7b3465a66d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.162ex; height:2.676ex;" alt="{\displaystyle A^{\mathrm {T} }}"></span> ottenuta scambiando ogni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb5a346f58c6568306a02596dd318d1b7e6b2c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.343ex;" alt="{\displaystyle a_{i,j}}"></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 a_{j,i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{j,i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76bb195d5314929dccc93b5b015dcffc15850e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.343ex;" alt="{\displaystyle a_{j,i}}"></span>, in altre parole ribaltando la matrice intorno alla sua diagonale principale. </p><p>Una matrice 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 a_{i,j}=a_{j,i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,j}=a_{j,i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2192031af3e1a9031b874fcb05bfe302a6b12a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.427ex; height:2.343ex;" alt="{\displaystyle a_{i,j}=a_{j,i}}"></span> è una <a href="/wiki/Matrice_simmetrica" title="Matrice simmetrica">matrice simmetrica</a>. In altre parole, <span class="mwe-math-element"><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> è simmetrica 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 A=A^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=A^{\mathrm {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170bdd9c2772de0b14fc9059fa9e66cfb7ef5553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.004ex; height:2.676ex;" alt="{\displaystyle A=A^{\mathrm {T} }}"></span>. Se tutti gli elementi che non stanno nella diagonale principale sono nulli, la matrice è detta <a href="/wiki/Matrice_diagonale" title="Matrice diagonale">diagonale</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Prodotto_di_matrici_quadrate">Prodotto di matrici quadrate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=14" title="Modifica la sezione Prodotto di matrici quadrate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=14" title="Edit section's source code: Prodotto di matrici quadrate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tra le più importanti matrici <span class="mwe-math-element"><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\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> vi è la <a href="/wiki/Matrice_identit%C3%A0" title="Matrice identità">matrice identità</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 I_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.242ex; height:2.509ex;" alt="{\displaystyle I_{n}}"></span>: si tratta di una matrice avente 1 su ogni elemento della diagonale e 0 altrove. La matrice è importante perché rappresenta l'<a href="/wiki/Elemento_neutro" title="Elemento neutro">elemento neutro</a> rispetto al prodotto: infatti le matrici <span class="mwe-math-element"><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\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> possono essere moltiplicate fra loro, e vale (oltre a quelle scritte sopra) la proprietà seguente per ogni <span class="mwe-math-element"><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>: </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 AI_{n}=I_{n}A=A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AI_{n}=I_{n}A=A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc36c4a5492f1d2033551d0abec92b832a1fd250" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.556ex; height:2.509ex;" alt="{\displaystyle AI_{n}=I_{n}A=A,}"></span></dd></dl> <p>ossia è l'elemento neutro del prodotto. Nello spazio delle matrici <span class="mwe-math-element"><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\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> sono quindi definiti una somma e un prodotto, e le proprietà elencate fin qui asseriscono che l'insieme è un <a href="/wiki/Anello_(algebra)" title="Anello (algebra)">anello</a>, simile all'anello dei <a href="/wiki/Numeri_interi" class="mw-redirect" title="Numeri interi">numeri interi</a>, con l'unica differenza che il prodotto di matrici non è commutativo. </p> <div class="mw-heading mw-heading3"><h3 id="Determinante">Determinante</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=15" title="Modifica la sezione Determinante" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=15" title="Edit section's source code: Determinante"><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/Determinante_(algebra)" title="Determinante (algebra)">Determinante (algebra)</a></b>.</span></div> </div> <p>Un'importante quantità definita a partire da 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> è il suo <a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">determinante</a>. 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 \det A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2d8fe180a2f848cf11e82a535b193cfe718742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.36ex; height:2.176ex;" alt="{\displaystyle \det A}"></span>, questo numero fornisce molte informazioni essenziali sulla matrice. Ad esempio, determina se la matrice è <a href="/wiki/Matrice_invertibile" title="Matrice invertibile">invertibile</a>, cioè se esiste una 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> 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 AB=BA=I_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mi>B</mi> <mi>A</mi> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB=BA=I_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/168794a263102ab93410d65c28bdc8bb955c8d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.1ex; height:2.509ex;" alt="{\displaystyle AB=BA=I_{n}.}"></span></dd></dl> <p>Il determinante è l'ingrediente fondamentale della <a href="/wiki/Regola_di_Cramer" title="Regola di Cramer">regola di Cramer</a>, utile a risolvere alcuni <a href="/wiki/Sistema_lineare" class="mw-redirect" title="Sistema lineare">sistemi lineari</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Polinomio_caratteristico,_autovettori,_diagonalizzabilità"><span id="Polinomio_caratteristico.2C_autovettori.2C_diagonalizzabilit.C3.A0"></span>Polinomio caratteristico, autovettori, diagonalizzabilità</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=16" title="Modifica la sezione Polinomio caratteristico, autovettori, diagonalizzabilità" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=16" title="Edit section's source code: Polinomio caratteristico, autovettori, 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/Autovettore_e_autovalore" title="Autovettore e autovalore">Autovettore e autovalore</a></b>, <b><a href="/wiki/Polinomio_caratteristico" title="Polinomio caratteristico">Polinomio caratteristico</a></b> e <b><a href="/wiki/Diagonalizzabilit%C3%A0" title="Diagonalizzabilità">Diagonalizzabilità</a></b>.</span></div> </div> <p>La traccia e il determinante possono essere racchiuse in un oggetto ancora più raffinato, di fondamentale importanza nello studio delle <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazioni lineari</a>: il <a href="/wiki/Polinomio_caratteristico" title="Polinomio caratteristico">polinomio caratteristico</a>, un <a href="/wiki/Polinomio" title="Polinomio">polinomio</a> le cui <a href="/wiki/Radice_(matematica)" title="Radice (matematica)">radici</a> sono gli <a href="/wiki/Autovettore_e_autovalore" title="Autovettore e autovalore">autovalori</a> della matrice. La conoscenza di autovalori e autovettori consente ad esempio di studiare la <a href="/wiki/Similitudine_fra_matrici" class="mw-redirect" title="Similitudine fra matrici">similitudine fra matrici</a>, in particolare la similitudine ad una matrice diagonale. </p> <div class="mw-heading mw-heading2"><h2 id="Classi_di_matrici_reali_e_complesse">Classi di matrici reali e complesse</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=17" title="Modifica la sezione Classi di matrici reali e complesse" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=17" title="Edit section's source code: Classi di matrici reali e complesse"><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/Glossario_sulle_matrici" title="Glossario sulle matrici">Glossario sulle matrici</a></b>.</span></div> </div> <p>Oltre alle matrici diagonali e simmetriche già introdotte, vi sono altre categorie di matrici importanti. </p> <ul><li>Le <a href="/wiki/Matrice_antisimmetrica" title="Matrice antisimmetrica">matrici antisimmetriche</a>, in cui i valori nelle caselle in posizioni simmetriche rispetto alla <i>diagonale principale</i> sono opposti: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,j}=-a_{j,i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,j}=-a_{j,i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6be8288773511b3633023ab1c96b332a4e189a8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.235ex; height:2.676ex;" alt="{\displaystyle a_{i,j}=-a_{j,i}}"></span>.</li> <li>Le <a href="/wiki/Matrice_hermitiana" title="Matrice hermitiana">matrici hermitiane</a> (o auto-aggiunte), in cui i valori nelle caselle di posizioni simmetriche rispetto alla diagonale principale sono <a href="/wiki/Complesso_coniugato" title="Complesso coniugato">complessi coniugati</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_{i,j}=a_{j,i}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,j}=a_{j,i}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c274d16bb24fb1ba3d82b499f0524f5ffdad0d33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.427ex; height:3.176ex;" alt="{\displaystyle a_{i,j}=a_{j,i}^{*}}"></span>.</li> <li>Un <a href="/wiki/Quadrato_magico" title="Quadrato magico">quadrato magico</a> è una matrice quadrata in cui la somma dei valori su ogni riga, colonna o diagonale è sempre la stessa.</li> <li>Le <a href="/wiki/Matrice_di_Toeplitz" title="Matrice di Toeplitz">matrici di Toeplitz</a> hanno valori costanti sulle diagonali parallele alla principale: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,j}=a_{i+1,j+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,j}=a_{i+1,j+1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c342192f7923ac5977ae1dc8c46e2043afaa1361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.275ex; height:2.343ex;" alt="{\displaystyle a_{i,j}=a_{i+1,j+1}.}"></span></li> <li>Le <a href="/wiki/Matrice_stocastica" title="Matrice stocastica">matrici stocastiche</a> sono matrici quadrate le cui colonne sono <a href="/w/index.php?title=Vettore_di_probabilit%C3%A0&action=edit&redlink=1" class="new" title="Vettore di probabilità (la pagina non esiste)">vettori di probabilità</a>, cioè sequenze di reali compresi tra 0 e 1 con somma uguale a 1; esse sono usate per definire le <a href="/wiki/Catena_di_Markov" class="mw-redirect" title="Catena di Markov">catene di Markov</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Spazio_di_matrici">Spazio di matrici</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=18" title="Modifica la sezione Spazio di matrici" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=18" title="Edit section's source code: Spazio di matrici"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lo spazio di tutte le matrici <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> a valori in un fissato <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> è indicato generalmente 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^{m\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{m\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a40b12a6c68466431891e4ad965fe811743d714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.343ex;" alt="{\displaystyle K^{m\times n}}"></span> o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(m,n,K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(m,n,K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993cb77b2aa9ac77464403e504e8a8f5139ff6e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.82ex; height:2.843ex;" alt="{\displaystyle M(m,n,K)}"></span>. Per quanto già visto, questo spazio è un gruppo abeliano con la somma. Considerato anche con la moltiplicazione per scalare, l'insieme ha una struttura di <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</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 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>. </p><p>Questo spazio ha una <a href="/wiki/Base_canonica" class="mw-redirect" title="Base canonica">base canonica</a>, composta da tutte le matrici <span class="mwe-math-element"><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_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c7a1c781b037551cd83fc9f1939d47f8a3cf4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.018ex; height:2.343ex;" alt="{\displaystyle e_{i,j}}"></span> aventi valore 1 sulla casella di posto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}"></span> e zero in tutte le altre. La base consta 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 mn}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mn}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348cd26a0b7a0034f57a951e2cf5f637dd47c1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:1.676ex;" alt="{\displaystyle mn}"></span> elementi, e quindi lo spazio <span class="mwe-math-element"><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^{m\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{m\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a40b12a6c68466431891e4ad965fe811743d714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.343ex;" alt="{\displaystyle K^{m\times n}}"></span> ha 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 mn}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mn}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348cd26a0b7a0034f57a951e2cf5f637dd47c1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:1.676ex;" alt="{\displaystyle mn}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Algebra_su_campo">Algebra su campo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=19" title="Modifica la sezione Algebra su campo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=19" title="Edit section's source code: Algebra su campo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nel caso <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c9d8e54796e7de7d4738510cc10bc3fc55d48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle m=n}"></span> delle matrici quadrate, è definito anche il prodotto. Con questa ulteriore operazione, lo spazio <span class="mwe-math-element"><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^{n\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{n\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10a4b78b51e6db58ee3f4b73b7e89eb589be5df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.577ex; height:2.343ex;" alt="{\displaystyle K^{n\times n}}"></span>, indicato anche 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 M(n,K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(n,K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b3d293c50c2eb7685534261e5445cdac13b5ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.746ex; height:2.843ex;" alt="{\displaystyle M(n,K)}"></span>, eredita una struttura di <a href="/wiki/Anello_con_unit%C3%A0" class="mw-redirect" title="Anello con unità">anello con unità</a>. Tale struttura è compatibile con quella di spazio vettoriale definita sopra, e fornisce quindi un esempio basilare di <a href="/wiki/Algebra_su_campo" title="Algebra su campo">algebra su campo</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizzazioni">Generalizzazioni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=20" title="Modifica la sezione Generalizzazioni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=20" title="Edit section's source code: Generalizzazioni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una <i>matrice infinita</i> può essere definita come una successione di elementi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb5a346f58c6568306a02596dd318d1b7e6b2c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.343ex;" alt="{\displaystyle a_{i,j}}"></span>, indicizzati da coppie di <a href="/wiki/Numeri_naturali" class="mw-redirect" title="Numeri naturali">numeri naturali</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 (i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}"></span>, senza nessun limite superiore per entrambi. </p><p>Più in generale, una generalizzazione del concetto di matrice è costruita prendendo due insiemi di indici <span class="mwe-math-element"><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,C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>,</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R,C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce992b91b5e18ba8547f8442fd23b57c5b7d654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.564ex; height:2.509ex;" alt="{\displaystyle R,C}"></span> qualsiasi (parametrizzanti le "righe" e le "colonne") e definendo una matrice come un'applicazione: </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\colon R\times C\rightarrow V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>R</mi> <mo>×</mo> <mi>C</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\colon R\times C\rightarrow V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27ad3afb8a63895dcaaaaf64aeffd38099887a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.549ex; height:2.176ex;" alt="{\displaystyle A\colon R\times C\rightarrow V}"></span></dd></dl> <p>a valori in un altro dato 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 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>. La matrice usuale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> corrisponde al 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 R=\{1,2,\ldots ,m\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\{1,2,\ldots ,m\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1feed32ae8e0409eb2cdac4afceda698ac3dabff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.765ex; height:2.843ex;" alt="{\displaystyle R=\{1,2,\ldots ,m\}}"></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 C=\{1,2,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\{1,2,\ldots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d857bb2f34392001756a8db2ea0d8bc3e4c4b6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.121ex; height:2.843ex;" alt="{\displaystyle C=\{1,2,\ldots ,n\}}"></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 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 ad esempio l'insieme dei numeri reali o complessi. </p><p>Questa definizione generale si serve solo di nozioni insiemistiche e non ricorre a nozioni visive e intuitive come quella di schieramento rettangolare. Consente di trattare casi molto generali: ad esempio matrici le cui righe e colonne sono etichettate da indici in un qualunque sottoinsieme <span class="mwe-math-element"><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> degli interi <span class="mwe-math-element"><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 {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>, matrici etichettate da coppie o in generale 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 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>-uple di interi come quelle che si incontrano nella <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">meccanica quantistica</a> o nella chimica molecolare, matrici infinite etichettate con gli insiemi <span class="mwe-math-element"><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 {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> come quelle che permettono di rappresentare <a href="/wiki/Successione_polinomiale" class="mw-redirect" title="Successione polinomiale">successioni polinomiali</a> o <a href="/wiki/Serie_formale" class="mw-redirect" title="Serie formale">serie formali</a> con due variabili. </p><p>Per poter definire somma, prodotto e altre operazioni sulle matrici, è opportuno che 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 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 dotato di tali operazioni, ad esempio che sia un <a href="/wiki/Anello_(algebra)" title="Anello (algebra)">anello</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Funzione_di_matrice">Funzione di matrice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice&veaction=edit&section=21" title="Modifica la sezione Funzione di matrice" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice&action=edit&section=21" title="Edit section's source code: Funzione di matrice"><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/Funzione_di_matrice" title="Funzione di matrice">Funzione di matrice</a></b>.</span></div> </div> <p>La teoria delle <a href="/wiki/Funzione_(matematica)" title="Funzione (matematica)">funzioni</a> di matrice è di grande interesse per lo studio dei <a href="/wiki/Sistemi_differenziali" class="mw-redirect" title="Sistemi differenziali">sistemi differenziali</a>: in generale la funzione di una matrice non coincide con la matrice delle funzioni dei suoi elementi, ma si dimostra sfruttando il <a href="/wiki/Teorema_di_Hamilton-Cayley" title="Teorema di Hamilton-Cayley">teorema di Hamilton-Cayley</a> che ciascun suo elemento è una <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> di queste ultime. </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=Matrice&veaction=edit&section=22" 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=Matrice&action=edit&section=22" 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"><ol class="references"> <li id="cite_note-mactutor-1"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-mactutor_1-0">a</a></i></sup> <sup><i><a href="#cite_ref-mactutor_1-1">b</a></i></sup> <sup><i><a href="#cite_ref-mactutor_1-2">c</a></i></sup> <sup><i><a href="#cite_ref-mactutor_1-3">d</a></i></sup> <sup><i><a href="#cite_ref-mactutor_1-4">e</a></i></sup> <sup><i><a href="#cite_ref-mactutor_1-5">f</a></i></sup> <sup><i><a href="#cite_ref-mactutor_1-6">g</a></i></sup> <sup><i><a href="#cite_ref-mactutor_1-7">h</a></i></sup> <sup><i><a href="#cite_ref-mactutor_1-8">i</a></i></sup> <sup><i><a href="#cite_ref-mactutor_1-9">j</a></i></sup></span> <span class="reference-text">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Matrices_and_determinants.html">Storia dell'uso delle matrici e dei determinanti</a> su <a href="/wiki/MacTutor" title="MacTutor">MacTutor</a></span> </li> <li id="cite_note-mactutor9-2"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-mactutor9_2-0">a</a></i></sup> <sup><i><a href="#cite_ref-mactutor9_2-1">b</a></i></sup></span> <span class="reference-text">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Nine_chapters.html">Il <i>Nove capitoli sulle arti matematiche</i></a> su <a href="/wiki/MacTutor" title="MacTutor">MacTutor</a></span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text">Il testo è consultabile on-line: <a rel="nofollow" class="external text" href="http://books.google.it/books?id=QoAHAAAAcAAJ&printsec=frontcover&source=gbs_navlinks_s#v=onepage&q=&f=false"><i>Treatise of Algebra</i></a>.</span> </li> <li id="cite_note-mactutorv-4"><a href="#cite_ref-mactutorv_4-0"><b>^</b></a> <span class="reference-text">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/Biographies/Vandermonde.html">Biografia di Vandermonde</a> su <a href="/wiki/MacTutor" title="MacTutor">MacTutor</a></span> </li> <li id="cite_note-5"><a href="#cite_ref-5"><b>^</b></a> <span class="reference-text">L'<i><a href="/wiki/Abstract" title="Abstract">abstract</a></i> del testo è consultabile on-line: <a rel="nofollow" class="external text" href="http://books.google.com/books?id=xOkAAAAAYAAJ&pg=PA100&ei=A6kkS5v5Lpe6yQS9kMyGCw&hl=it&cd=1#v=onepage&q=&f=false"><i>Memoir on the theory of matrices</i> in <i>Proceedings of the Royal Society of London, Volume 9</i></a>.</span> </li> <li id="cite_note-6"><a href="#cite_ref-6"><b>^</b></a> <span class="reference-text">S. C. Althoen and R. McLaughlin, "Gauss-Jordan Reduction: A Brief History," American Mathematical Monthly, 94:130–142 (1987).</span> </li> <li id="cite_note-bro89-1-7"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-bro89-1_7-0">a</a></i></sup> <sup><i><a href="#cite_ref-bro89-1_7-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFBronson_1989">Bronson 1989</a>, <i>Preface</i></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=Matrice&veaction=edit&section=23" 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=Matrice&action=edit&section=23" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFBronson_1989" class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Richard Bronson, <span style="font-style:italic;">Schaum's Outline of Theory and Problems of Matrix Operations</span>, New York, McGraw-Hill, 1989, pp. 230 pagine., <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-07-007978-1" title="Speciale:RicercaISBN/0-07-007978-1">0-07-007978-1</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Fulvio Bisi, Francesco Bonsante e Sonia Brivio, <span style="font-style:italic;">3</span>, in <span style="font-style:italic;">Lezioni di algebra lineare con applicazioni alla geometria analitica</span>, Pavia, La Dotta, agosto 2013, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-98648-02-2" title="Speciale:RicercaISBN/88-98648-02-2">88-98648-02-2</a>.</cite></li> <li><cite id="CITEREFBUR05" class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) David M. Burton, <span style="font-style:italic;">The History of Mathematics: An Introduction</span>, 6ª edizione, McGraw-Hill, 1º dicembre 2005, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-07-110635-1" title="Speciale:RicercaISBN/978-0-07-110635-1">978-0-07-110635-1</a>.</cite></li> <li><cite id="CITEREFFEL62" class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Richard W. Jr. Feldmann, <span style="font-style:italic;">Arthur Cayley - Founder of Matrix Theory</span>, The Mathematics Teacher, 55, 1962, Pagine 482-484..</cite></li> <li><cite id="CITEREFGolub" class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a href="/wiki/Gene_H._Golub" title="Gene H. Golub">Gene H. Golub</a>, Charles F. Van Loan, <span style="font-style:italic;">Matrix computations</span>, 3ª edizione, 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></ul> <p><br /> </p> <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=Matrice&veaction=edit&section=24" 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=Matrice&action=edit&section=24" 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/Autovettore_e_autovalore" title="Autovettore e autovalore">Autovettore e autovalore</a></li> <li><a href="/wiki/Determinante_(algebra)" title="Determinante (algebra)">Determinante</a></li> <li><a href="/wiki/Glossario_sulle_matrici" title="Glossario sulle matrici">Glossario sulle matrici</a></li> <li><a href="/wiki/Matrice_associata_ad_una_trasformazione_lineare" class="mw-redirect" title="Matrice associata ad una trasformazione lineare">Matrice associata ad una trasformazione lineare</a></li> <li><a href="/wiki/Norma_matriciale" title="Norma matriciale">Norma matriciale</a></li> <li><a href="/wiki/Rango_(algebra_lineare)" title="Rango (algebra lineare)">Rango (algebra lineare)</a></li> <li><a href="/wiki/Sistema_lineare" class="mw-redirect" title="Sistema lineare">Sistema lineare</a></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=Matrice&veaction=edit&section=25" 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=Matrice&action=edit&section=25" 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/matrice" class="extiw" title="wikt:matrice">Wikizionario</a></li> <li class="" title=""><a href="https://it.wikiversity.org/wiki/Matrice" class="extiw" title="v:Matrice">Wikiversità</a></li> <li class="" title=""><span class="plainlinks" title="commons:Category:Matrices"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Matrices?uselang=it">Wikimedia Commons</a></span></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/matrice" class="extiw" title="wikt:matrice">matrice</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 sulla <b><a href="https://it.wikiversity.org/wiki/Matrice" class="extiw" title="v:Matrice">matrice</a></b></li> <li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file sulla <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Matrices?uselang=it">matrice</a></span></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=Matrice&veaction=edit&section=26" 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=Matrice&action=edit&section=26" 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="CITEREFSapere.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.sapere.it/enciclopedia/matrice.html"><span style="font-style:italic;">matrice</span></a>, su <span style="font-style:italic;">sapere.it</span>, <a href="/wiki/De_Agostini" title="De Agostini">De Agostini</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q44337#P6706" 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="CITEREFEnciclopedia_della_Matematica" class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/matrice_(Enciclopedia-della-Matematica)/"><span style="font-style:italic;">matrice</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/Q44337#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="CITEREFBritannica.com" 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://www.britannica.com/topic/matrix-mathematics"><span style="font-style:italic;">matrix</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q44337#P1417" 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="CITEREFOpen_Library" 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://openlibrary.org/subjects/matrix"><span style="font-style:italic;">Opere riguardanti Matrix</span></a>, su <span style="font-style:italic;"><a href="/wiki/Open_Library" class="mw-redirect" title="Open Library">Open Library</a></span>, <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q44337#P3847" 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="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Matrix.html"><span style="font-style:italic;">Matrix</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q44337#P2812" 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/Matrix"><span style="font-style:italic;">Matrix</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/Q44337#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> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="italiano">IT</abbr></span>) <a rel="nofollow" class="external text" href="http://simplemath.online/calcolatrici-online/matrici-e-vettori.html">Calcolatrice per matrici e vettori online</a></li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Matrices_and_determinants.html">Storia dell'uso delle matrici e dei determinanti</a> su <a href="/wiki/MacTutor" title="MacTutor">MacTutor</a></li> <li>(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://planetmath.org/encyclopedia/Matrix.html">Matrice</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100615101722/http://planetmath.org/encyclopedia/Matrix.html">Archiviato</a> il 15 giugno 2010 in <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>. su <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a></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"><tbody><tr><th colspan="3" style="background:#ffc0cb;"><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" title="Template:Algebra"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Algebra&action=edit&redlink=1" class="new" title="Discussioni template:Algebra (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&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" title="Algebra">Algebra</a></span></th></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Numero" title="Numero">Numeri</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Numero_naturale" title="Numero naturale">Naturali</a><b> ·</b> <a href="/wiki/Numero_intero" title="Numero intero">Interi</a><b> ·</b> <a href="/wiki/Numero_razionale" title="Numero razionale">Razionali</a><b> ·</b> <a href="/wiki/Numero_irrazionale" title="Numero irrazionale">Irrazionali</a><b> ·</b> <a href="/wiki/Numero_algebrico" title="Numero algebrico">Algebrici</a><b> ·</b> <a href="/wiki/Numero_trascendente" title="Numero trascendente">Trascendenti</a><b> ·</b> <a href="/wiki/Numero_reale" title="Numero reale">Reali</a><b> ·</b> <a href="/wiki/Numero_complesso" title="Numero complesso">Complessi</a><b> ·</b> <a href="/wiki/Numero_ipercomplesso" title="Numero ipercomplesso">Numero ipercomplesso</a><b> ·</b> <a href="/wiki/Numero_p-adico" title="Numero p-adico">Numero p-adico</a><b> ·</b> <a href="/wiki/Numero_duale" title="Numero duale">Duali</a><b> ·</b> <a href="/wiki/Numero_complesso_iperbolico" title="Numero complesso iperbolico">Complessi iperbolici</a></td><td rowspan="10" class="navbox_image"><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/58px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="58" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/87px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, 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colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Algebra_elementare" title="Algebra elementare">Algebra elementare</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Equazione" title="Equazione">Equazione</a><b> ·</b> <a href="/wiki/Disequazione" title="Disequazione">Disequazione</a><b> ·</b> <a href="/wiki/Polinomio" title="Polinomio">Polinomio</a><b> ·</b> <a href="/wiki/Triangolo_di_Tartaglia" title="Triangolo di Tartaglia">Triangolo di Tartaglia</a><b> ·</b> <a href="/wiki/Teorema_binomiale" title="Teorema binomiale">Teorema binomiale</a><b> ·</b> <a href="/wiki/Teorema_del_resto" title="Teorema del resto">Teorema del resto</a><b> ·</b> <a href="/wiki/Lemma_di_Gauss_(polinomi)" title="Lemma di Gauss (polinomi)">Lemma di Gauss</a><b> ·</b> <a href="/wiki/Teorema_delle_radici_razionali" title="Teorema delle radici razionali">Teorema delle radici razionali</a><b> ·</b> <a href="/wiki/Regola_di_Ruffini" title="Regola di Ruffini">Regola di Ruffini</a><b> ·</b> <a href="/wiki/Criterio_di_Eisenstein" title="Criterio di Eisenstein">Criterio di Eisenstein</a><b> ·</b> <a href="/wiki/Criterio_di_Cartesio" title="Criterio di Cartesio">Criterio di Cartesio</a><b> ·</b> <a href="/wiki/Disequazione_con_il_valore_assoluto" title="Disequazione con il valore assoluto">Disequazione con il valore assoluto</a><b> ·</b> <a href="/wiki/Segno_(matematica)" title="Segno (matematica)">Segno</a><b> ·</b> <a href="/wiki/Metodo_di_Gauss-Seidel" title="Metodo di Gauss-Seidel">Metodo di Gauss-Seidel</a><b> ·</b> <a href="/wiki/Polinomio_simmetrico" title="Polinomio simmetrico">Polinomio simmetrico</a><b> ·</b> <a href="/wiki/Funzione_simmetrica" title="Funzione simmetrica">Funzione simmetrica</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Elementi di <a href="/wiki/Calcolo_combinatorio" title="Calcolo combinatorio">Calcolo combinatorio</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Fattoriale" title="Fattoriale">Fattoriale</a><b> ·</b> <a href="/wiki/Permutazione" title="Permutazione">Permutazione</a><b> ·</b> <a href="/wiki/Disposizione" title="Disposizione">Disposizione</a><b> ·</b> <a href="/wiki/Combinazione" title="Combinazione">Combinazione</a><b> ·</b> <a href="/wiki/Dismutazione_(matematica)" title="Dismutazione (matematica)">Dismutazione</a><b> ·</b> <a href="/wiki/Principio_di_inclusione-esclusione" title="Principio di inclusione-esclusione">Principio di inclusione-esclusione</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Concetti fondamentali di <a href="/wiki/Teoria_dei_numeri" title="Teoria dei numeri">Teoria dei numeri</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Primi</th><td colspan="1"><a href="/wiki/Numero_primo" title="Numero primo">Numero primo</a><b> ·</b> <a href="/wiki/Teorema_dell%27infinit%C3%A0_dei_numeri_primi" title="Teorema dell'infinità dei numeri primi">Teorema dell'infinità dei numeri primi</a><b> ·</b> <a href="/wiki/Crivello_di_Eratostene" title="Crivello di Eratostene">Crivello di Eratostene</a><b> ·</b> <a href="/wiki/Crivello_di_Atkin" title="Crivello di Atkin">Crivello di Atkin</a><b> ·</b> <a href="/wiki/Test_di_primalit%C3%A0" title="Test di primalità">Test di primalità</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_dell%27aritmetica" title="Teorema fondamentale dell'aritmetica">Teorema fondamentale dell'aritmetica</a></td></tr><tr><th class="subnavbox_group">Divisori</th><td colspan="1"><a href="/wiki/Interi_coprimi" title="Interi coprimi">Interi coprimi</a><b> ·</b> <a href="/wiki/Identit%C3%A0_di_B%C3%A9zout" title="Identità di Bézout">Identità di Bézout</a><b> ·</b> <a href="/wiki/Massimo_comun_divisore" title="Massimo comun divisore">MCD</a><b> ·</b> <a href="/wiki/Minimo_comune_multiplo" title="Minimo comune multiplo">mcm</a><b> ·</b> <a href="/wiki/Algoritmo_di_Euclide" title="Algoritmo di Euclide">Algoritmo di Euclide</a><b> ·</b> <a href="/wiki/Algoritmo_esteso_di_Euclide" title="Algoritmo esteso di Euclide">Algoritmo esteso di Euclide</a><b> ·</b> <a href="/wiki/Criteri_di_divisibilit%C3%A0" title="Criteri di divisibilità">Criteri di divisibilità</a><b> ·</b> <a href="/wiki/Divisore" title="Divisore">Divisore</a></td></tr><tr><th class="subnavbox_group"><a href="/wiki/Aritmetica_modulare" title="Aritmetica modulare">Aritmetica modulare</a></th><td colspan="1"><a href="/wiki/Teorema_cinese_del_resto" title="Teorema cinese del resto">Teorema cinese del resto</a><b> ·</b> <a href="/wiki/Piccolo_teorema_di_Fermat" title="Piccolo teorema di Fermat">Piccolo teorema di Fermat</a><b> ·</b> <a href="/wiki/Teorema_di_Eulero_(aritmetica_modulare)" title="Teorema di Eulero (aritmetica modulare)">Teorema di Eulero</a><b> ·</b> <a href="/wiki/Funzione_%CF%86_di_Eulero" title="Funzione φ di Eulero">Funzione φ di Eulero</a><b> ·</b> <a href="/wiki/Teorema_di_Wilson" title="Teorema di Wilson">Teorema di Wilson</a><b> ·</b> <a href="/wiki/Reciprocit%C3%A0_quadratica" title="Reciprocità quadratica">Reciprocità quadratica</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Teoria_dei_gruppi" title="Teoria dei gruppi">Teoria dei gruppi</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><table class="subnavbox"><tbody><tr><th class="subnavbox_group">Gruppi</th><td colspan="1"><a href="/wiki/Gruppo_(matematica)" title="Gruppo (matematica)">Gruppo</a> (<a href="/wiki/Gruppo_finito" title="Gruppo finito">finito</a><b> ·</b> <a href="/wiki/Gruppo_ciclico" title="Gruppo ciclico">ciclico</a><b> ·</b> <a href="/wiki/Gruppo_abeliano" title="Gruppo abeliano">abeliano</a>)<b> ·</b> <a href="/wiki/Gruppo_primario" title="Gruppo primario">Gruppo primario</a><b> ·</b> <a href="/wiki/Gruppo_quoziente" title="Gruppo quoziente">Gruppo quoziente</a><b> ·</b> <a href="/wiki/Gruppo_nilpotente" title="Gruppo nilpotente">Gruppo nilpotente</a><b> ·</b> <a href="/wiki/Gruppo_risolubile" title="Gruppo risolubile">Gruppo risolubile</a><b> ·</b> <a href="/wiki/Gruppo_simmetrico" title="Gruppo simmetrico">Gruppo simmetrico</a><b> ·</b> <a href="/wiki/Gruppo_diedrale" title="Gruppo diedrale">Gruppo diedrale</a><b> ·</b> <a href="/wiki/Gruppo_semplice" title="Gruppo semplice">Gruppo semplice</a><b> ·</b> <a href="/wiki/Gruppo_sporadico" title="Gruppo sporadico">Gruppo sporadico</a><b> ·</b> <a href="/wiki/Gruppo_mostro" title="Gruppo mostro">Gruppo mostro</a><b> ·</b> <a href="/wiki/Gruppo_di_Klein" title="Gruppo di Klein">Gruppo di Klein</a><b> ·</b> <a href="/wiki/Gruppo_dei_quaternioni" title="Gruppo dei quaternioni">Gruppo dei quaternioni</a><b> ·</b> <a href="/wiki/Gruppo_generale_lineare" title="Gruppo generale lineare">Gruppo generale lineare</a><b> ·</b> <a href="/wiki/Gruppo_ortogonale" title="Gruppo ortogonale">Gruppo ortogonale</a><b> ·</b> <a href="/wiki/Gruppo_unitario" title="Gruppo unitario">Gruppo unitario</a><b> ·</b> <a href="/wiki/Gruppo_unitario_speciale" title="Gruppo unitario speciale">Gruppo unitario speciale</a><b> ·</b> <a href="/wiki/Gruppo_residualmente_finito" title="Gruppo residualmente finito">Gruppo residualmente finito</a><b> ·</b> <a href="/wiki/Gruppo_spaziale" title="Gruppo spaziale">Gruppo spaziale</a><b> ·</b> <a href="/wiki/Gruppo_profinito" title="Gruppo profinito">Gruppo profinito</a><b> ·</b> <a href="/wiki/Out(Fn)" title="Out(Fn)">Out(F<sub>n</sub>)</a><b> ·</b> <a href="/wiki/Parola_(teoria_dei_gruppi)" title="Parola (teoria dei gruppi)">Parola</a><b> ·</b> <a href="/wiki/Prodotto_diretto" title="Prodotto diretto">Prodotto diretto</a><b> ·</b> <a href="/wiki/Prodotto_semidiretto" title="Prodotto semidiretto">Prodotto semidiretto</a><b> ·</b> <a href="/wiki/Prodotto_intrecciato" title="Prodotto intrecciato">Prodotto intrecciato</a></td></tr><tr><th class="subnavbox_group">Teoremi</th><td colspan="1"><a href="/wiki/Alternativa_di_Tits" title="Alternativa di Tits">Alternativa di Tits</a><b> ·</b> <a href="/wiki/Teorema_di_isomorfismo" title="Teorema di isomorfismo">Teorema di isomorfismo</a><b> ·</b> <a href="/wiki/Teorema_di_Lagrange_(teoria_dei_gruppi)" title="Teorema di Lagrange (teoria dei gruppi)">Teorema di Lagrange</a><b> ·</b> <a href="/wiki/Teorema_di_Cauchy_(teoria_dei_gruppi)" title="Teorema di Cauchy (teoria dei gruppi)">Teorema di Cauchy</a><b> ·</b> <a href="/wiki/Teoremi_di_Sylow" title="Teoremi di Sylow">Teoremi di Sylow</a><b> ·</b> <a href="/wiki/Teorema_di_Cayley" title="Teorema di Cayley">Teorema di Cayley</a><b> ·</b> <a href="/wiki/Gruppo_abeliano#Classificazione" title="Gruppo abeliano">Teorema di struttura dei gruppi abeliani finiti</a><b> ·</b> <a href="/wiki/Lemma_della_farfalla" title="Lemma della farfalla">Lemma della farfalla</a><b> ·</b> <a href="/wiki/Lemma_del_ping-pong" title="Lemma del ping-pong">Lemma del ping-pong</a><b> ·</b> <a href="/wiki/Classificazione_dei_gruppi_semplici_finiti" title="Classificazione dei gruppi semplici finiti">Classificazione dei gruppi semplici finiti</a></td></tr><tr><th class="subnavbox_group">Sottoinsiemi</th><td colspan="1"><a href="/wiki/Sottogruppo" title="Sottogruppo">Sottogruppo</a><b> ·</b> <a href="/wiki/Sottogruppo_normale" title="Sottogruppo normale">Sottogruppo normale</a><b> ·</b> <a href="/wiki/Sottogruppo_caratteristico" title="Sottogruppo caratteristico">Sottogruppo caratteristico</a><b> ·</b> <a href="/wiki/Sottogruppo_di_Frattini" title="Sottogruppo di Frattini">Sottogruppo di Frattini</a><b> ·</b> <a href="/wiki/Sottogruppo_di_torsione" title="Sottogruppo di torsione">Sottogruppo di torsione</a><b> ·</b> <a href="/wiki/Classe_laterale" title="Classe laterale">Classe laterale</a><b> ·</b> <a href="/wiki/Classe_di_coniugio" title="Classe di coniugio">Classe di coniugio</a><b> ·</b> <a href="/wiki/Serie_di_composizione" title="Serie di composizione">Serie di composizione</a></td></tr><tr><td colspan="2" class="navbox_center"><a href="/wiki/Omomorfismo_di_gruppi" title="Omomorfismo di gruppi">Omomorfismo</a><b> ·</b> <a href="/wiki/Isomorfismo_tra_gruppi" title="Isomorfismo tra gruppi">Isomorfismo</a><b> ·</b> <a href="/wiki/Automorfismo_interno" title="Automorfismo interno">Automorfismo interno</a><b> ·</b> <a href="/wiki/Automorfismo_esterno" title="Automorfismo esterno">Automorfismo esterno</a><b> ·</b> <a href="/wiki/Permutazione" title="Permutazione">Permutazione</a><b> ·</b> <a href="/wiki/Presentazione_di_un_gruppo" title="Presentazione di un gruppo">Presentazione di un gruppo</a><b> ·</b> <a href="/wiki/Azione_di_gruppo" title="Azione di gruppo">Azione di gruppo</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Teoria_degli_anelli" title="Teoria degli anelli">Teoria degli anelli</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Anello_(algebra)" title="Anello (algebra)">Anello</a> (<a href="/wiki/Anello_artiniano" title="Anello artiniano">artiniano</a><b> ·</b> <a href="/wiki/Anello_noetheriano" title="Anello noetheriano">noetheriano</a><b> ·</b> <a href="/wiki/Anello_locale" title="Anello locale">locale</a>)<b> ·</b> <a href="/wiki/Caratteristica_(algebra)" title="Caratteristica (algebra)">Caratteristica</a><b> ·</b> <a href="/wiki/Ideale_(matematica)" title="Ideale (matematica)">Ideale</a> (<a href="/wiki/Ideale_primo" title="Ideale primo">primo</a><b> ·</b> <a href="/wiki/Ideale_massimale" title="Ideale massimale">massimale</a>)<b> ·</b> <a href="/wiki/Dominio_d%27integrit%C3%A0" title="Dominio d'integrità">Dominio</a> (<a href="/wiki/Dominio_a_fattorizzazione_unica" title="Dominio a fattorizzazione unica">a fattorizzazione unica</a><b> ·</b> <a href="/wiki/Dominio_ad_ideali_principali" title="Dominio ad ideali principali">a ideali principali</a><b> ·</b> <a href="/wiki/Dominio_euclideo" title="Dominio euclideo">euclideo</a>)<b> ·</b> <a class="mw-selflink selflink">Matrice</a><b> ·</b> <a href="/wiki/Anello_semplice" title="Anello semplice">Anello semplice</a><b> ·</b> <a href="/wiki/Anello_degli_endomorfismi" title="Anello degli endomorfismi">Anello degli endomorfismi</a><b> ·</b> <a href="/wiki/Teorema_di_Artin-Wedderburn" title="Teorema di Artin-Wedderburn">Teorema di Artin-Wedderburn</a><b> ·</b> <a href="/wiki/Modulo_(algebra)" title="Modulo (algebra)">Modulo</a><b> ·</b> <a href="/wiki/Dominio_di_Dedekind" title="Dominio di Dedekind">Dominio di Dedekind</a><b> ·</b> <a href="/wiki/Estensione_di_anelli" title="Estensione di anelli">Estensione di anelli</a><b> ·</b> <a href="/wiki/Teorema_della_base_di_Hilbert" title="Teorema della base di Hilbert">Teorema della base di Hilbert</a><b> ·</b> <a href="/wiki/Anello_di_Gorenstein" title="Anello di Gorenstein">Anello di Gorenstein</a><b> ·</b> <a href="/wiki/Base_di_Gr%C3%B6bner" title="Base di Gröbner">Base di Gröbner</a><b> ·</b> <a href="/wiki/Prodotto_tensoriale" title="Prodotto tensoriale">Prodotto tensoriale</a><b> ·</b> <a href="/wiki/Primo_associato" title="Primo associato">Primo associato</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;"><a href="/wiki/Teoria_dei_campi_(matematica)" class="mw-redirect" title="Teoria dei campi (matematica)">Teoria dei campi</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><table class="subnavbox"><tbody><tr><td colspan="2" class="navbox_center"><a href="/wiki/Campo_(matematica)" title="Campo (matematica)">Campo</a><b> ·</b> <a href="/wiki/Polinomio_irriducibile" title="Polinomio irriducibile">Polinomio irriducibile</a><b> ·</b> <a href="/wiki/Polinomio_ciclotomico" title="Polinomio ciclotomico">Polinomio ciclotomico</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_dell%27algebra" title="Teorema fondamentale dell'algebra">Teorema fondamentale dell'algebra</a><b> ·</b> <a href="/wiki/Campo_finito" title="Campo finito">Campo finito</a><b> ·</b> <a href="/wiki/Automorfismo" title="Automorfismo">Automorfismo</a><b> ·</b> <a href="/wiki/Endomorfismo_di_Frobenius" title="Endomorfismo di Frobenius">Endomorfismo di Frobenius</a></td></tr><tr><th class="subnavbox_group">Estensioni</th><td colspan="1"><a href="/wiki/Campo_di_spezzamento" title="Campo di spezzamento">Campo di spezzamento</a><b> ·</b> <a href="/wiki/Estensione_di_campi" title="Estensione di campi">Estensione di campi</a><b> ·</b> <a href="/wiki/Estensione_algebrica" title="Estensione algebrica">Estensione algebrica</a><b> ·</b> <a href="/wiki/Estensione_separabile" title="Estensione separabile">Estensione separabile</a><b> ·</b> <a href="/wiki/Chiusura_algebrica" title="Chiusura algebrica">Chiusura algebrica</a><b> ·</b> <a href="/wiki/Campo_di_numeri" title="Campo di numeri">Campo di numeri</a><b> ·</b> <a href="/wiki/Estensione_normale" title="Estensione normale">Estensione normale</a><b> ·</b> <a href="/wiki/Estensione_di_Galois" title="Estensione di Galois">Estensione di Galois</a><b> ·</b> <a href="/wiki/Estensione_abeliana" title="Estensione abeliana">Estensione abeliana</a><b> ·</b> <a href="/wiki/Estensione_ciclotomica" title="Estensione ciclotomica">Estensione ciclotomica</a><b> ·</b> <a href="/wiki/Teoria_di_Kummer" title="Teoria di Kummer">Teoria di Kummer</a></td></tr><tr><th class="subnavbox_group">Teoria di Galois</th><td colspan="1"><a href="/wiki/Gruppo_di_Galois" title="Gruppo di Galois">Gruppo di Galois</a><b> ·</b> <a href="/wiki/Teoria_di_Galois" title="Teoria di Galois">Teoria di Galois</a><b> ·</b> <a href="/wiki/Teorema_fondamentale_della_teoria_di_Galois" title="Teorema fondamentale della teoria di Galois">Teorema fondamentale della teoria di Galois</a><b> ·</b> <a href="/wiki/Teorema_di_Abel-Ruffini" title="Teorema di Abel-Ruffini">Teorema di Abel-Ruffini</a><b> ·</b> <a href="/wiki/Costruzioni_con_riga_e_compasso" title="Costruzioni con riga e compasso">Costruzioni con riga e compasso</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">Altre <a href="/wiki/Struttura_algebrica" title="Struttura algebrica">strutture algebriche</a></th><td colspan="1" class="navbox_list navbox_odd" style="text-align:left;"><a href="/wiki/Magma_(matematica)" title="Magma (matematica)">Magma</a><b> ·</b> <a href="/wiki/Semigruppo" title="Semigruppo">Semigruppo</a><b> ·</b> <a href="/wiki/Corpo_(matematica)" title="Corpo (matematica)">Corpo</a><b> ·</b> <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">Spazio vettoriale</a><b> ·</b> <a href="/wiki/Algebra_su_campo" title="Algebra su campo">Algebra su campo</a><b> ·</b> <a href="/wiki/Algebra_di_Lie" title="Algebra di Lie">Algebra di Lie</a><b> ·</b> <a href="/wiki/Algebra_differenziale" title="Algebra differenziale">Algebra differenziale</a><b> ·</b> <a href="/wiki/Algebra_di_Clifford" title="Algebra di Clifford">Algebra di Clifford</a><b> ·</b> <a href="/wiki/Gruppo_topologico" title="Gruppo topologico">Gruppo topologico</a><b> ·</b> <a href="/wiki/Gruppo_ordinato" title="Gruppo ordinato">Gruppo ordinato</a><b> ·</b> <a href="/wiki/Quasi-anello" title="Quasi-anello">Quasi-anello</a><b> ·</b> <a href="/wiki/Algebra_di_Boole" title="Algebra di Boole">Algebra di Boole</a></td></tr><tr><th colspan="1" class="navbox_group" style="background:#FFE0E0; text-align:right;">argomenti</th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Teoria_delle_categorie" title="Teoria delle categorie">Teoria delle categorie</a><b> ·</b> <a href="/wiki/Algebra_lineare" title="Algebra lineare">Algebra lineare</a><b> ·</b> <a href="/wiki/Algebra_commutativa" title="Algebra commutativa">Algebra commutativa</a><b> ·</b> <a href="/wiki/Algebra_omologica" title="Algebra omologica">Algebra omologica</a><b> ·</b> <a href="/wiki/Algebra_astratta" title="Algebra astratta">Algebra astratta</a><b> ·</b> <a href="/wiki/Algebra_computazionale" class="mw-redirect" title="Algebra computazionale">Algebra computazionale</a><b> ·</b> <a href="/wiki/Algebra_differenziale" title="Algebra differenziale">Algebra differenziale</a><b> ·</b> <a href="/wiki/Algebra_universale" title="Algebra universale">Algebra universale</a></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r141815314"><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 class="mw-selflink selflink">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 href="/wiki/Autovettore_e_autovalore" title="Autovettore e autovalore">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; text-align:right;"><a href="/wiki/Prodotto_scalare" title="Prodotto scalare">Prodotto scalare</a></th><td colspan="1" class="navbox_list navbox_even" style="text-align:left;"><a href="/wiki/Forma_bilineare" title="Forma bilineare">Forma bilineare</a><b> ·</b> <a href="/wiki/Sottospazio_ortogonale" title="Sottospazio ortogonale">Sottospazio ortogonale</a><b> ·</b> <a href="/wiki/Spazio_euclideo" title="Spazio euclideo">Spazio euclideo</a><b> ·</b> <a href="/wiki/Base_ortonormale" title="Base ortonormale">Base ortonormale</a><b> ·</b> <a href="/wiki/Algoritmo_di_Lagrange" title="Algoritmo di Lagrange">Algoritmo di Lagrange</a><b> ·</b> <a href="/wiki/Segnatura_(algebra_lineare)" title="Segnatura (algebra lineare)">Segnatura</a><b> ·</b> <a href="/wiki/Teorema_di_Sylvester" title="Teorema di Sylvester">Teorema di Sylvester</a><b> ·</b> <a href="/wiki/Ortogonalizzazione_di_Gram-Schmidt" title="Ortogonalizzazione di Gram-Schmidt">Gram-Schmidt</a><b> ·</b> <a href="/wiki/Forma_sesquilineare" title="Forma sesquilineare">Forma sesquilineare</a><b> ·</b> <a href="/wiki/Forma_sesquilineare#Forma_hermitiana" title="Forma sesquilineare">Forma hermitiana</a><b> ·</b> <a href="/wiki/Teorema_spettrale" title="Teorema spettrale">Teorema spettrale</a></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Nuovo_soggettario" title="Nuovo soggettario">Thesaurus BNCF</a> <span class="uid"><a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=17734">17734</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85082210">sh85082210</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4037968-1">4037968-1</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Spagna" title="Biblioteca nazionale di Spagna">BNE</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="spagnolo">ES</abbr></span>) <a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX529678">XX529678</a> <a rel="nofollow" class="external text" href="http://datos.bne.es/resource/XX529678">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119324420">cb119324420</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119324420">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007557992905171">987007557992905171</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></a></span> <b><a href="/wiki/Portale:Matematica" title="Portale:Matematica">Portale Matematica</a></b>: accedi alle voci di Wikipedia che trattano di matematica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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Matrice - Wikipedia