{"id":315,"date":"2023-07-06T10:41:54","date_gmt":"2023-07-06T10:41:54","guid":{"rendered":"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/"},"modified":"2023-07-06T10:41:54","modified_gmt":"2023-07-06T10:41:54","slug":"conjugado-de-matriz-complexa-e-conjugado-transposto","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/","title":{"rendered":"Matriz complexa, conjugada e transposta"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que s\u00e3o matrizes complexas, matrizes conjugadas e matrizes transpostas conjugadas. Agora eles se parecem muito com voc\u00ea, mas voc\u00ea ver\u00e1 como no final da p\u00e1gina entender\u00e1 perfeitamente a diferen\u00e7a entre cada um. Al\u00e9m disso, veremos exemplos de cada tipo e suas propriedades.<\/p>\n<h2 class=\"wp-block-heading\"> matriz complexa<\/h2>\n<p> Antes de ver a explica\u00e7\u00e3o da matriz conjugada e da matriz conjugada transposta, vamos revisar o conceito de matriz complexa:<\/p>\n<h3 class=\"wp-block-heading\"> O que \u00e9 uma matriz complexa?<\/h3>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Uma <strong>matriz complexa<\/strong> \u00e9 uma matriz que possui um determinado n\u00famero complexo entre seus elementos. <\/p>\n<div style=\"background-color:#fffde7;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p style=\"text-align:left\"> Lembre-se que <strong>um n\u00famero complexo ou imagin\u00e1rio<\/strong> \u00e9 um n\u00famero composto por uma parte real e uma parte imagin\u00e1ria, indicada pela letra i. Por exemplo:<\/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-0fee5c908395e5217468d53482c2e362_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3+5i\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<\/div>\n<p> .<\/p>\n<h3 class=\"wp-block-heading\"> Exemplos de matrizes complexas<\/h3>\n<p> Vejamos alguns exemplos de matrizes multidimensionais complexas:<\/p>\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <span style=\"text-decoration: underline;\">Exemplo de matriz complexa de ordem 2 \u00d7 2<\/span> <\/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\/matrice-complexe-de-dimension-22152-1.webp\" alt=\"matriz complexa de dimens\u00e3o 2x2\" class=\"wp-image-2087\" width=\"154\" height=\"67\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <span style=\"text-decoration: underline;\">Exemplo de uma matriz complexa de dimens\u00e3o 3\u00d73<\/span> <\/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\/matrice-complexe-de-dimension-32153-1.webp\" alt=\"matriz complexa de dimens\u00e3o 3x3\" class=\"wp-image-2088\" width=\"247\" height=\"105\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <span style=\"text-decoration: underline;\">Exemplo de uma matriz complexa de tamanho 4\u00d74<\/span> <\/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\/matrice-complexe-de-dimension-42154-2.webp\" alt=\"matriz complexa de dimens\u00e3o 4x4\" class=\"wp-image-2090\" width=\"325\" height=\"133\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"> matriz conjugada<\/h2>\n<p> Depois de vermos qual \u00e9 a defini\u00e7\u00e3o de uma matriz complexa, vamos ver o que s\u00e3o uma matriz conjugada e uma matriz conjugada transposta:<\/p>\n<h3 class=\"wp-block-heading\"> O que \u00e9 uma matriz conjugada?<\/h3>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Uma <strong>matriz conjugada<\/strong> \u00e9 uma matriz complexa em que todos os seus elementos foram substitu\u00eddos pelos seus conjugados, ou seja, o sinal da parte imagin\u00e1ria de todos os seus n\u00fameros complexos foi alterado.<\/p>\n<p> A matriz conjugada de<\/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> \u00e9 expresso por uma barra horizontal acima:<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ed46621c8e86e9928a3cf0a340bfa33d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\overline{A}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo de uma matriz conjugada <\/h3>\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-conjuguee-en-3-dimensions-3-2.webp\" alt=\"exemplo de conjugado de matriz, como conjugar uma matriz\" class=\"wp-image-2105\" width=\"566\" height=\"102\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h3 class=\"wp-block-heading\"> Propriedades da matriz conjugada<\/h3>\n<p> As caracter\u00edsticas deste tipo de matriz s\u00e3o as seguintes:<\/p>\n<ul>\n<li> O <span style=\"color:#1976d2;\"><strong>conjugado<\/strong><\/span> de uma matriz conjugada \u00e9 a matriz original.<\/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-3f90bce927a6e9e5f862c7a3b7030f27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\overline{\\bigl( \\ \\overline{A} \\vphantom{A^{9^1}} \\ \\bigr)} = A\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"78\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> <span style=\"color:#1976d2;\"><strong>Adicionar (ou subtrair) duas matrizes<\/strong><\/span> e conjugar o resultado \u00e9 o mesmo que primeiro conjugar as duas matrizes separadamente e depois adicion\u00e1-las (ou subtra\u00ed-las).<\/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-e67c3d39dcf5be98c4a0eb655a3e286b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\overline{\\bigl( A \\pm B \\bigr)} = \\overline{A} \\pm \\overline{B}\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"139\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> <span style=\"color:#1976d2;\"><strong>O produto conjugado<\/strong><\/span> de duas matrizes \u00e9 igual a conjugar as duas matrizes separadamente e depois calcular a multiplica\u00e7\u00e3o da matriz.<\/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-0245e2cb0bbffcefc968f5018aeefab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\overline{\\bigl( A \\cdot B \\bigr)} = \\overline{A} \\cdot \\overline{B}\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"121\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> <span style=\"color:#1976d2;\"><strong>Multiplicar uma matriz por um escalar<\/strong><\/span> e conjugar o resultado \u00e9 o mesmo que primeiro fazer os conjugados do escalar e da matriz e depois resolver o produto.<\/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-e416d234161773402aa585c1b97db935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\overline{\\bigl( k \\cdot A \\bigr)} = \\overline{k} \\cdot \\overline{A}\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"112\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> <span style=\"color:#1976d2;\"><strong>Transpor uma matriz<\/strong><\/span> e depois conjug\u00e1-la significa primeiro conjugar a matriz e depois transp\u00f4-la.<\/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-08b80e702d649410b95f31576b828248_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\overline{\\bigl(A^t \\bigr)} = \\left( \\overline{A}\\right)^t\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"94\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> Fazer o <span style=\"color:#1976d2;\"><strong>inverso de uma matriz<\/strong><\/span> e depois conjug\u00e1-la \u00e9 o mesmo que conjugar a matriz e depois invert\u00ea-la.<\/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-565953dab3b0d2eaef6ea45d9e50743e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\overline{\\bigl( A^{-1} \\bigr)} = \\left(\\overline{A} \\right)^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"118\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> A <span style=\"color:#1976d2;\"><strong>classifica\u00e7\u00e3o de uma matriz conjugada<\/strong><\/span> \u00e9 igual \u00e0 classifica\u00e7\u00e3o da mesma matriz n\u00e3o conjugada.<\/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-a63700aed5a47afbab56ddf555872c84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle rg\\left(\\overline{A}\\right) =rg(A)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"117\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> \u00c9 indiferente calcular o <span style=\"color:#1976d2;\"><strong>tra\u00e7o de uma matriz conjugada<\/strong><\/span> ou calcular o tra\u00e7o da mesma matriz sem conjuga\u00e7\u00e3o e depois conjugar o resultado.<\/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-c2a59bd3f66ffd209062d6e33364c0aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  tr\\left(\\overline{A}\\right) =\\overline{tr(A)}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"113\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> Por fim, tomar o <span style=\"color:#1976d2;\"><strong>determinante de uma matriz conjugada<\/strong><\/span> equivale a calcular o conjugado do resultado do determinante da mesma matriz sem conjuga\u00e7\u00e3o.<\/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-535d01ae156409915068c6a0b88c88c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle det\\left(\\overline{A}\\right) = \\overline{det(A)}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"131\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Matriz transposta conjugada<\/h2>\n<p> Por fim, depois de ver como conjugar uma matriz, vamos passar ao conceito de matriz transposta conjugada:<\/p>\n<h3 class=\"wp-block-heading\"> O que \u00e9 uma matriz transposta (ou transposta) conjugada?<\/h3>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> A <strong>matriz conjugada transposta (ou transposta)<\/strong> \u00e9 aquela obtida ap\u00f3s ter transposto uma matriz e feito seu conjugado.<\/p>\n<p> Este tipo de matriz tamb\u00e9m \u00e9 chamada de matriz adjunta ou simplesmente matriz adjunta. Al\u00e9m disso, geralmente \u00e9 representado por um asterisco<\/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-c05635b1a884461b4fa83975f0ca7d37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(A^*)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"33\" style=\"vertical-align: -5px;\"><\/p>\n<p> , embora existam matem\u00e1ticos que o desenham como<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0d4c81a666954cf4d9d7889c69274641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^*\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<p> qualquer<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-baf599f65a55110769b27904ec93bc83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^H\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"25\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo de matriz de transposi\u00e7\u00e3o conjugada<\/h3>\n<p> Aqui est\u00e1 um exemplo de c\u00e1lculo da transposta (ou transposta conjugada) de uma 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-d4fc37df446a4600709c54e8b1b78072_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}1+3i&amp;2-i &amp; -4i \\\\[1.1ex] 6 &amp; 8+2i &amp; 3-5i \\\\[1.1ex] 7i &amp; 1+9i &amp; -2+i\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"237\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Primeiro transpomos a matriz A:<\/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-fdc97bb655f8e2f18abbc2e6d480c02b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^t=\\begin{pmatrix}1+3i&amp; 6 &amp; 7i \\\\[1.1ex] 2-i &amp; 8+2i &amp; 1+9i \\\\[1.1ex] -4i &amp; 3-5i &amp; -2+i\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"243\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E ent\u00e3o calculamos a matriz conjugada da transposta, ou seja, mudamos o sinal da parte imagin\u00e1ria de todos os n\u00fameros complexos:<\/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-209bcf07d842e6157663ddc03909d544_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*=\\overline{A^t}=\\begin{pmatrix}1-3i&amp; 6 &amp; -7i \\\\[1.1ex] 2+i &amp; 8-2i &amp; 1-9i \\\\[1.1ex] 4i &amp; 3+5i &amp; -2-i\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"288\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, o resumo do c\u00e1lculo da matriz transposta conjugada \u00e9: <\/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\/matrice-de-transposee-conjuguee-a-3-215-3-1-dimensions.webp\" alt=\"matriz transposta conjugada de dimens\u00e3o 3x3\" class=\"wp-image-2167\" width=\"632\" height=\"107\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Propriedades da matriz transposta conjugada<\/h3>\n<p> As propriedades deste tipo de matriz quadrada s\u00e3o as seguintes:<\/p>\n<ul>\n<li> A <span style=\"color:#1976d2;\"><strong>matriz transposta conjugada<\/strong><\/span> de uma matriz previamente transposta e conjugada \u00e9 a matriz original.<\/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-ef2b419c47cda5f2bfe0fb1ab98f9791_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bigl(A^*\\bigr) ^* = A\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"80\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> A <span style=\"color:#1976d2;\"><strong>propriedade de adi\u00e7\u00e3o<\/strong><\/span> de matrizes transpostas conjugadas afirma que adicionar (ou subtrair) duas matrizes e depois aplicar esta opera\u00e7\u00e3o ao resultado \u00e9 equivalente a primeiro fazer a transposta conjugada de cada matriz e depois adicionar (ou subtrair) os resultados.<\/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-8a8c5ade3065db20fdd63c44a34a283d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bigl( A\\pm B \\bigr)^* = A^*\\pm B^*\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"158\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> <span><strong style=\"color: rgb(25, 118, 210);\">Multiplicar<\/strong><\/span> duas matrizes e depois fazer sua transposta conjugada d\u00e1 o mesmo resultado que o produto inverso das matrizes transpostas conjugadas.<\/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-211055521d5259b6b0e1edd88e14da87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bigl( A\\cdot B \\bigr)^* = B^*\\cdot A^*\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"140\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> Calcular a matriz transposta conjugada do <span style=\"color:#1976d2;\"><strong>produto de um escalar e uma matriz<\/strong><\/span> \u00e9 o mesmo que conjugar o n\u00famero complexo e encontrar a transposta conjugada da matriz separadamente e depois multiplicar.<\/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-56cd508ac5a34671995c3a158d0b64a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bigl( k\\cdot A \\bigr)^* = \\overline{k}\\cdot A^*\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"123\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> Se a matriz for invert\u00edvel, a ordem em que <span style=\"color:#1976d2;\"><strong>as opera\u00e7\u00f5es de invers\u00e3o da matriz e de transposi\u00e7\u00e3o conjugada<\/strong><\/span> s\u00e3o realizadas \u00e9 irrelevante.<\/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-f8b076e5f3f0d3c0ca92c5fc999550cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bigl( A^{-1} \\bigr)^*= \\bigl( A^* \\bigr)^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"132\" style=\"vertical-align: -7px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que s\u00e3o matrizes complexas, matrizes conjugadas e matrizes transpostas conjugadas. Agora eles se parecem muito com voc\u00ea, mas voc\u00ea ver\u00e1 como no final da p\u00e1gina entender\u00e1 perfeitamente a diferen\u00e7a entre cada um. Al\u00e9m disso, veremos exemplos de cada tipo e suas propriedades. matriz complexa Antes de ver a explica\u00e7\u00e3o da &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/\"> <span class=\"screen-reader-text\">Matriz complexa, conjugada e transposta<\/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-315","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 complexa, conjugada e transposta - 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\/conjugado-de-matriz-complexa-e-conjugado-transposto\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriz complexa, conjugada e transposta - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea ver\u00e1 o que s\u00e3o matrizes complexas, matrizes conjugadas e matrizes transpostas conjugadas. 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Al\u00e9m disso, veremos exemplos de cada tipo e suas propriedades. matriz complexa Antes de ver a explica\u00e7\u00e3o da &hellip; Matriz complexa, conjugada e transposta Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T10:41:54+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0fee5c908395e5217468d53482c2e362_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=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Matriz complexa, conjugada e transposta\",\"datePublished\":\"2023-07-06T10:41:54+00:00\",\"dateModified\":\"2023-07-06T10:41:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/\"},\"wordCount\":729,\"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\/conjugado-de-matriz-complexa-e-conjugado-transposto\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/\",\"url\":\"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/\",\"name\":\"Matriz complexa, conjugada e transposta - 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Agora eles se parecem muito com voc\u00ea, mas voc\u00ea ver\u00e1 como no final da p\u00e1gina entender\u00e1 perfeitamente a diferen\u00e7a entre cada um. Al\u00e9m disso, veremos exemplos de cada tipo e suas propriedades. matriz complexa Antes de ver a explica\u00e7\u00e3o da &hellip; Matriz complexa, conjugada e transposta Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/","article_published_time":"2023-07-06T10:41:54+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0fee5c908395e5217468d53482c2e362_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"4 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Matriz complexa, conjugada e transposta","datePublished":"2023-07-06T10:41:54+00:00","dateModified":"2023-07-06T10:41:54+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/"},"wordCount":729,"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\/conjugado-de-matriz-complexa-e-conjugado-transposto\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/","url":"https:\/\/mathority.org\/pt\/conjugado-de-matriz-complexa-e-conjugado-transposto\/","name":"Matriz complexa, conjugada e transposta - 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