{"id":324,"date":"2023-07-06T08:09:51","date_gmt":"2023-07-06T08:09:51","guid":{"rendered":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/"},"modified":"2023-07-06T08:09:51","modified_gmt":"2023-07-06T08:09:51","slug":"exemplos-de-matrizes-comutaveis-comutam","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/","title":{"rendered":"Matrizes comut\u00e1veis"},"content":{"rendered":"<p>Nesta p\u00e1gina explicamos o que s\u00e3o matrizes comut\u00e1veis. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos para entender bem o conceito e, por fim, encontrar\u00e1 um exerc\u00edcio resolvido passo a passo no qual aprendemos a calcular todas as matrizes que comutam com qualquer matriz.<\/p>\n<h2 class=\"wp-block-heading\"> O que s\u00e3o matrizes comut\u00e1veis? <\/h2>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p style=\"text-align:left\"> Duas <strong>matrizes s\u00e3o comut\u00e1veis<\/strong> se o resultado do seu produto n\u00e3o depender da ordem de multiplica\u00e7\u00e3o. Em outras palavras, matrizes comut\u00e1veis satisfazem a seguinte condi\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6eb88f6da7a7fb25d88ae172483b637c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot B = B \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"104\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<p> Esta \u00e9 a defini\u00e7\u00e3o de matrizes comut\u00e1veis, agora vamos ver um exemplo:<\/p>\n<h2 class=\"wp-block-heading\"> Exemplo de matrizes comut\u00e1veis<\/h2>\n<p> As duas matrizes a seguir de dimens\u00e3o 2\u00d72 podem ser alternadas entre elas:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4afa74407be7cf7a0142ce931dbba98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 0\\\\[1.1ex] 1 &amp; -1 \\end{pmatrix} \\quad B= \\begin{pmatrix} 3&amp; 0\\\\[1.1ex] 1 &amp; 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"229\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A comutabilidade das duas matrizes poderia ser demonstrada calculando seu produto em ambas as dire\u00e7\u00f5es: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-matrices-commutables-22152-1.webp\" alt=\"exemplo de matrizes comut\u00e1veis de dimens\u00e3o 2x2\" class=\"wp-image-2759\" width=\"377\" height=\"169\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Como voc\u00ea pode ver, o resultado de ambas as multiplica\u00e7\u00f5es \u00e9 o mesmo, independentemente da ordem em que s\u00e3o multiplicadas. Ent\u00e3o as matrizes<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> eles s\u00e3o comut\u00e1veis.<\/p>\n<h2 class=\"wp-block-heading\"> Exerc\u00edcio de troca de matriz resolvido<\/h2>\n<p> A seguir veremos passo a passo como resolver um exerc\u00edcio de matriz comut\u00e1vel:<\/p>\n<ul>\n<li> Determine todas as matrizes que comutam com a seguinte matriz quadrada:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0f69e9df9aa524aeabcc1716a92b5e8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; 1\\\\[1.1ex] 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Para resolver este problema criaremos uma matriz desconhecida:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee9183823ea39248018c37cbac3bf2ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B=\\begin{pmatrix} a &amp; b\\\\[1.1ex] c &amp; d \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"97\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Devemos, portanto, encontrar esta matriz desconhecida.<\/p>\n<p> Para fazer isso, aproveitaremos a propriedade que todas as matrizes comutadas satisfazem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6eb88f6da7a7fb25d88ae172483b637c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot B = B \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"104\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-98ac92178351b7dc235918b2bc02ed90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}3 &amp; 1\\\\[1.1ex] 1 &amp; 0\\end{pmatrix}\\cdot \\begin{pmatrix} a &amp; b\\\\[1.1ex] c &amp; d \\end{pmatrix} = \\begin{pmatrix} a &amp; b\\\\[1.1ex] c &amp; d \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 1\\\\[1.1ex] 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"291\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Agora multiplicamos as matrizes em ambos os lados da equa\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5bd3e34eadc944aa1aea8f323f9796ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 3a+c &amp;3b+d\\\\[1.1ex] a &amp; b \\end{pmatrix} =  \\begin{pmatrix}3a+b &amp; a\\\\[1.1ex] 3c+d &amp; c \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"256\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, para que a igualdade seja v\u00e1lida, as seguintes equa\u00e7\u00f5es devem ser atendidas:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d1f3094807b37f4fbc9875b5dddc5f25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{l} 3a+c=3a+b \\\\[2ex] 3b+d=a \\\\[2ex] a=3c+d\\\\[2ex] b= c \\end{array}\\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"140\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o tudo o que precisamos fazer \u00e9 resolver o sistema de equa\u00e7\u00f5es. Da \u00faltima equa\u00e7\u00e3o podemos deduzir que<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> deve ser igual a<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"c\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-08cf30e231c919c278f8af358e06df4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=c\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E se estas duas inc\u00f3gnitas forem equivalentes, a terceira equa\u00e7\u00e3o se repete com a segunda, podemos portanto elimin\u00e1-la:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b3e25af3ab248d099ae0515f9912cdf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{l} 3a+c=3a+b \\\\[2ex] 3b+d=a \\\\[2ex] \\cancel{a=3c+d}\\\\[2ex] b= c \\end{array}\\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"140\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Al\u00e9m disso, da primeira equa\u00e7\u00e3o n\u00e3o podemos tirar quaisquer conclus\u00f5es, porque: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7608b6e0c6fb91990657aa470c48f5f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3a+c=3a+b \\ \\xrightarrow{b \\ = \\ c} \\ 3a+b=3a+b\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"305\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-762f5011b3045c2f20b77035ae8473f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3a=3a\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"60\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-378ae3997c3bacdce60cc73eb967fdb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a=a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, ficamos apenas com a segunda e \u00faltima equa\u00e7\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3486d0076e11ddae06ffbfcbb3fab66a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{l} 3b+d=a \\\\[2ex] b= c \\end{array}\\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"101\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Para que as matrizes comutando com a matriz<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> s\u00e3o todos aqueles que verificam as duas equa\u00e7\u00f5es anteriores. Portanto, substituindo as express\u00f5es encontradas na matriz desconhecida desde o in\u00edcio, podemos encontrar a forma das matrizes que comutam com<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-944477c7f7578892a57aa3b7c7dd8268_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A:\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ccd60f786e1324e748a7d91e41f86442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} a &amp; b\\\\[1.1ex] c &amp; d \\end{pmatrix} \\ \\longrightarrow \\ \\begin{pmatrix} 3b+d &amp; b \\\\[1.1ex] b &amp; d \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"206\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ouro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e8716946f6a868f015e0d62f28bc540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> s\u00e3o dois n\u00fameros reais.<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Ent\u00e3o, um exemplo de uma matriz que comutaria com a matriz<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> seria o seguinte:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0c22c13d155ba46f6a9d0f6891747699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 6 &amp; 1 \\\\[1.1ex] 1 &amp; 3  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"54\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Propriedades de matrizes comut\u00e1veis<\/h2>\n<p> Matrizes comut\u00e1veis possuem as seguintes caracter\u00edsticas:<\/p>\n<ul>\n<li> Matrizes comut\u00e1veis <strong>n\u00e3o possuem a propriedade transitiva<\/strong> . Em outras palavras, mesmo que a matriz\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> comutar com matrizes<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> , isso n\u00e3o significa que<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> s\u00e3o comut\u00e1veis entre eles.<\/li>\n<\/ul>\n<ul>\n<li> <a href=\"https:\/\/mathority.org\/pt\/matriz-diagonal\/\">Matrizes diagonais<\/a> comutam entre si, ou seja, uma matriz diagonal comuta com qualquer outra matriz diagonal.<\/li>\n<\/ul>\n<ul>\n<li> Da mesma forma, uma matriz escalar comuta igualmente com todas as matrizes. Por exemplo, a <a href=\"https:\/\/mathority.org\/pt\">matriz Identidade ou Unidade<\/a> comuta com todas as matrizes.<\/li>\n<\/ul>\n<ul>\n<li> Duas <a href=\"https:\/\/mathority.org\/pt\/matriz-hermitiana-ou-hermitiana\/\">matrizes Hermitianas<\/a> comutam se seus autovetores (ou autovetores) coincidirem.<\/li>\n<\/ul>\n<ul>\n<li> Obviamente, a <a href=\"https:\/\/mathority.org\/pt\/matriz-nula-zero\/\">matriz zero<\/a> tamb\u00e9m comuta com todas as matrizes.<\/li>\n<\/ul>\n<ul>\n<li> Se o produto de duas <a href=\"https:\/\/mathority.org\/pt\/exemplos-e-propriedades-de-matrizes-simetricas\/\">matrizes sim\u00e9tricas<\/a> d\u00e1 outra matriz sim\u00e9trica, ent\u00e3o as duas matrizes devem comutar.<\/li>\n<\/ul>\n<ul>\n<li> Se a diagonaliza\u00e7\u00e3o de duas matrizes puder ser feita simultaneamente, elas dever\u00e3o ser comut\u00e1veis. Portanto, essas duas matrizes tamb\u00e9m compartilham a mesma base ortonormal de autovetores ou autovetores.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina explicamos o que s\u00e3o matrizes comut\u00e1veis. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos para entender bem o conceito e, por fim, encontrar\u00e1 um exerc\u00edcio resolvido passo a passo no qual aprendemos a calcular todas as matrizes que comutam com qualquer matriz. O que s\u00e3o matrizes comut\u00e1veis? Duas matrizes s\u00e3o comut\u00e1veis se o resultado do &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/\"> <span class=\"screen-reader-text\">Matrizes comut\u00e1veis<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[37],"tags":[],"class_list":["post-324","post","type-post","status-publish","format-standard","hentry","category-tipos-de-tabelas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matrizes comut\u00e1veis - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matrizes comut\u00e1veis - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina explicamos o que s\u00e3o matrizes comut\u00e1veis. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos para entender bem o conceito e, por fim, encontrar\u00e1 um exerc\u00edcio resolvido passo a passo no qual aprendemos a calcular todas as matrizes que comutam com qualquer matriz. O que s\u00e3o matrizes comut\u00e1veis? Duas matrizes s\u00e3o comut\u00e1veis se o resultado do &hellip; Matrizes comut\u00e1veis Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T08:09:51+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6eb88f6da7a7fb25d88ae172483b637c_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Matrizes comut\u00e1veis\",\"datePublished\":\"2023-07-06T08:09:51+00:00\",\"dateModified\":\"2023-07-06T08:09:51+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/\"},\"wordCount\":543,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Tipos de tabelas\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/\",\"url\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/\",\"name\":\"Matrizes comut\u00e1veis - Mathority\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-07-06T08:09:51+00:00\",\"dateModified\":\"2023-07-06T08:09:51+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Matrizes comut\u00e1veis\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/pt\/#website\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"name\":\"Mathority\",\"description\":\"Onde a curiosidade encontra o c\u00e1lculo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/pt\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\",\"name\":\"Equipe Mathoridade\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Equipe Mathoridade\"},\"sameAs\":[\"http:\/\/mathority.org\/pt\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Matrizes comut\u00e1veis - Mathority","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/","og_locale":"pt_BR","og_type":"article","og_title":"Matrizes comut\u00e1veis - Mathority","og_description":"Nesta p\u00e1gina explicamos o que s\u00e3o matrizes comut\u00e1veis. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos para entender bem o conceito e, por fim, encontrar\u00e1 um exerc\u00edcio resolvido passo a passo no qual aprendemos a calcular todas as matrizes que comutam com qualquer matriz. O que s\u00e3o matrizes comut\u00e1veis? Duas matrizes s\u00e3o comut\u00e1veis se o resultado do &hellip; Matrizes comut\u00e1veis Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/","article_published_time":"2023-07-06T08:09:51+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6eb88f6da7a7fb25d88ae172483b637c_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"3 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Matrizes comut\u00e1veis","datePublished":"2023-07-06T08:09:51+00:00","dateModified":"2023-07-06T08:09:51+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/"},"wordCount":543,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Tipos de tabelas"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/","url":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/","name":"Matrizes comut\u00e1veis - Mathority","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-07-06T08:09:51+00:00","dateModified":"2023-07-06T08:09:51+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-comutaveis-comutam\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Matrizes comut\u00e1veis"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/pt\/#website","url":"https:\/\/mathority.org\/pt\/","name":"Mathority","description":"Onde a curiosidade encontra o c\u00e1lculo!","publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/pt\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/mathority.org\/pt\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/pt\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00","name":"Equipe Mathoridade","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Equipe Mathoridade"},"sameAs":["http:\/\/mathority.org\/pt"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/324","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/comments?post=324"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/324\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=324"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=324"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=324"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}