{"id":318,"date":"2023-07-06T09:59:20","date_gmt":"2023-07-06T09:59:20","guid":{"rendered":"https:\/\/mathority.org\/pt\/matriz-unitaria\/"},"modified":"2023-07-06T09:59:20","modified_gmt":"2023-07-06T09:59:20","slug":"matriz-unitaria","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/matriz-unitaria\/","title":{"rendered":"Matriz unit\u00e1ria"},"content":{"rendered":"<p>Nesta p\u00e1gina explicamos o que \u00e9 a matriz unit\u00e1ria e, al\u00e9m disso, ilustramos com diversos exerc\u00edcios para que seja bem compreendida. Voc\u00ea tamb\u00e9m descobrir\u00e1 quais s\u00e3o todas as propriedades desse tipo de matriz t\u00e3o importantes para a \u00e1lgebra linear.<\/p>\n<h2 class=\"wp-block-heading\"> O que \u00e9 uma matriz unit\u00e1ria?<\/h2>\n<p> A defini\u00e7\u00e3o de matriz unit\u00e1ria \u00e9 a seguinte: <\/p>\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\"> Uma <strong>matriz unit\u00e1ria<\/strong> \u00e9 uma matriz complexa que multiplicada por sua <a href=\"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/\">matriz transposta conjugada<\/a> \u00e9 igual \u00e0 matriz identidade. Ou seja, a seguinte condi\u00e7\u00e3o \u00e9 atendida:<\/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-4b382ce59eebf001ec9f1a07cebfd7d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"U\\cdot U^* = U^* \\cdot U =I\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"152\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> 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-2b60fc262803f27ba3717d8ec4eb656d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"U\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 uma matriz unit\u00e1ria e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db11158221d0c663d1a4da78e077a3f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"U^*\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<p> sua transposta conjugada.<\/p>\n<\/div>\n<p> Portanto, esta condi\u00e7\u00e3o implica que <strong>a inversa de uma matriz unit\u00e1ria \u00e9 a sua transposta conjugada<\/strong> , pois, segundo a defini\u00e7\u00e3o de uma matriz inversa, uma matriz \u00e9 a inversa de outra se o seu produto for equivalente \u00e0 matriz d&#8217;identificar .<\/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-d8f035ef94e00b67acffd2881944642f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{c} U \\cdot U^*  =I \\\\[2ex] U \\cdot U^{-1} = I\\end{array} \\right\\} \\longrightarrow \\ U^*=U^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"236\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, uma matriz unit\u00e1ria ser\u00e1 sempre uma <strong>matriz regular ou n\u00e3o degenerada<\/strong> , pois sempre ter\u00e1 uma inversa.<\/p>\n<p> Por outro lado, o an\u00e1logo de uma matriz unit\u00e1ria num ambiente de n\u00fameros reais \u00e9 a <strong>matriz ortogonal<\/strong> , e neste caso \u00e9 verdade que a matriz unit\u00e1ria multiplicada pela sua transposta \u00e9 igual \u00e0 matriz identidade.<\/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-0cc2f9e0f4e1c41a0e3ab402c6465d59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"U\\cdot U^t = U^t \\cdot U =I\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"149\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, neste caso a matriz inversa de U seria diretamente a sua matriz transposta (ou transposta).<\/p>\n<h2 class=\"wp-block-heading\"> Exemplos de matrizes unit\u00e1rias<\/h2>\n<h3 class=\"wp-block-heading\"> Exemplo de uma matriz unit\u00e1ria de dimens\u00e3o 2\u00d72<\/h3>\n<p> Depois de vermos o conceito de matriz unit\u00e1ria, veremos um exemplo de matriz unit\u00e1ria 2\u00d72 para entend\u00ea-la bem: <\/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-matrice-unitaire-de-dimension-22152-1.webp\" alt=\"exemplo de matriz unit\u00e1ria de dimens\u00e3o 2x2\" class=\"wp-image-2204\" width=\"222\" height=\"69\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Esta matriz \u00e9 unit\u00e1ria porque a multiplica\u00e7\u00e3o dela mesma pela sua matriz conjugada d\u00e1 a matriz Identidade (ou Unidade):<\/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-11df575022f8a50881fedc994f4f12af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle U\\cdot U^*=\\cfrac{1}{3} \\begin{pmatrix} 2 &amp; -2+i \\\\[1.1ex] 2+i &amp; 2 \\end{pmatrix}\\cdot \\cfrac{1}{3} \\begin{pmatrix} 2 &amp; 2-i \\\\[1.1ex] -2-i &amp; 2 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"456\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E, como vimos anteriormente, qualquer matriz unit\u00e1ria \u00e9 comut\u00e1vel com sua transposta conjugada: <\/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-4848f3eab836be0996049e221bb8a8c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle U^*\\cdot U=\\cfrac{1}{3} \\begin{pmatrix} 2 &amp; 2-i \\\\[1.1ex] -2-i &amp; 2 \\end{pmatrix}\\cdot \\cfrac{1}{3} \\begin{pmatrix} 2 &amp; -2+i \\\\[1.1ex] 2+i &amp; 2 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"456\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Exemplo de matriz diagonal unit\u00e1ria<\/h3>\n<p> A <a href=\"https:\/\/mathority.org\/pt\/matriz-diagonal\/\">matriz diagonal<\/a> composta apenas pelo n\u00famero complexo <em>i<\/em> tamb\u00e9m \u00e9 um exemplo de matriz unit\u00e1ria, independente da dimens\u00e3o da matriz. Abaixo voc\u00ea tem um exerc\u00edcio resolvido que ilustra isso com uma matriz unit\u00e1ria de dimens\u00e3o 3 \u00d7 3: <\/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-matrice-unitaire-de-3-dimensions-32153-1.webp\" alt=\"exemplo de matriz unit\u00e1ria de dimens\u00e3o 3x3\" class=\"wp-image-2211\" width=\"153\" height=\"105\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Observe que se resolvermos o produto da matriz por sua transposta conjugada, isso d\u00e1 a matriz Identidade como solu\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-61bc73f95b9c2515595fe3ed2e18df3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle U\\cdot U^* =\\begin{pmatrix} i &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; i &amp; 0 \\\\[1.1ex] 0&amp; 0 &amp; i \\end{pmatrix}\\cdot \\begin{pmatrix} -i &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; -i &amp; 0 \\\\[1.1ex] 0&amp; 0 &amp; -i \\end{pmatrix}=\\begin{pmatrix} 1 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 1 &amp; 0 \\\\[1.1ex] 0&amp; 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"406\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E a mesma coisa acontece se multiplicarmos as matrizes ao contr\u00e1rio:<\/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-5cdf7b15d442ec89fde613ba2fd3fe45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle U^*\\cdot U =\\begin{pmatrix} -i &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; -i &amp; 0 \\\\[1.1ex] 0&amp; 0 &amp; -i \\end{pmatrix}\\cdot \\begin{pmatrix} i &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; i &amp; 0 \\\\[1.1ex] 0&amp; 0 &amp; i \\end{pmatrix}=\\begin{pmatrix} 1 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 1 &amp; 0 \\\\[1.1ex] 0&amp; 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"406\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A caracter\u00edstica desta matriz \u00e9 que ela serve como exemplo de matriz unit\u00e1ria de qualquer dimens\u00e3o, pois cada vez que a matriz \u00e9 formada pelo n\u00famero imagin\u00e1rio <em>i<\/em> na diagonal principal e os demais elementos s\u00e3o zero (0 ) ser\u00e1 uma matriz unit\u00e1ria.<\/p>\n<h2 class=\"wp-block-heading\"> Propriedades de uma matriz unit\u00e1ria<\/h2>\n<p> As propriedades das matrizes unit\u00e1rias s\u00e3o as seguintes:<\/p>\n<ul>\n<li> Obviamente, qualquer matriz unit\u00e1ria \u00e9 uma <a href=\"https:\/\/mathority.org\/pt\/matriz-normal\/\">matriz normal<\/a> . Embora nem todas as matrizes normais sejam matrizes unit\u00e1rias.<\/li>\n<\/ul>\n<ul>\n<li> Matrizes unit\u00e1rias s\u00e3o sempre <a href=\"https:\/\/mathority.org\/pt\/matriz-quadrada\/\">matrizes quadradas<\/a> .<\/li>\n<\/ul>\n<ul>\n<li> Todas as matrizes unit\u00e1rias s\u00e3o diagonaliz\u00e1veis, ou seja, podem ser transformadas em matrizes diagonais.<\/li>\n<\/ul>\n<ul>\n<li> O valor absoluto do determinante de uma matriz unit\u00e1ria \u00e9 sempre igual a 1.<\/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-3fd4555e54680e0af331c0cc03865df1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} det(U) \\end{vmatrix} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"93\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> A <a href=\"https:\/\/mathority.org\/pt\">matriz id\u00eantica<\/a> \u00e9 uma matriz unit\u00e1ria.<\/li>\n<\/ul>\n<ul>\n<li> para todos\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> , o conjunto de todas as matrizes unit\u00e1rias<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b31629cb1899edcc0029f841492d4f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\\times n\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> com a opera\u00e7\u00e3o de produto matricial, eles formam um grupo, denominado grupo de unidades.<\/li>\n<\/ul>\n<ul>\n<li> Portanto, a multiplica\u00e7\u00e3o de duas matrizes unit\u00e1rias da mesma ordem d\u00e1 outra matriz unit\u00e1ria.<\/li>\n<\/ul>\n<ul>\n<li> O m\u00f3dulo de todos os autovalores (ou autovalores) de uma matriz unit\u00e1ria \u00e9 sempre igual a 1.<\/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-05a9b3dcd4707885f2c0a2de613cfb57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} \\lambda \\end{vmatrix} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> Os autoespa\u00e7os deste tipo de matriz s\u00e3o ortogonais.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina explicamos o que \u00e9 a matriz unit\u00e1ria e, al\u00e9m disso, ilustramos com diversos exerc\u00edcios para que seja bem compreendida. Voc\u00ea tamb\u00e9m descobrir\u00e1 quais s\u00e3o todas as propriedades desse tipo de matriz t\u00e3o importantes para a \u00e1lgebra linear. O que \u00e9 uma matriz unit\u00e1ria? A defini\u00e7\u00e3o de matriz unit\u00e1ria \u00e9 a seguinte: Uma matriz &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/matriz-unitaria\/\"> <span class=\"screen-reader-text\">Matriz unit\u00e1ria<\/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-318","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>Matriz unit\u00e1ria - Matoridade<\/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\/matriz-unitaria\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriz unit\u00e1ria - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina explicamos o que \u00e9 a matriz unit\u00e1ria e, al\u00e9m disso, ilustramos com diversos exerc\u00edcios para que seja bem compreendida. Voc\u00ea tamb\u00e9m descobrir\u00e1 quais s\u00e3o todas as propriedades desse tipo de matriz t\u00e3o importantes para a \u00e1lgebra linear. O que \u00e9 uma matriz unit\u00e1ria? 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Voc\u00ea tamb\u00e9m descobrir\u00e1 quais s\u00e3o todas as propriedades desse tipo de matriz t\u00e3o importantes para a \u00e1lgebra linear. O que \u00e9 uma matriz unit\u00e1ria? 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