{"id":314,"date":"2023-07-06T10:46:31","date_gmt":"2023-07-06T10:46:31","guid":{"rendered":"https:\/\/mathority.org\/pt\/matriz-normal\/"},"modified":"2023-07-06T10:46:31","modified_gmt":"2023-07-06T10:46:31","slug":"matriz-normal","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/matriz-normal\/","title":{"rendered":"Matriz regular"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 uma matriz normal, bem como exemplos de matrizes normais. Al\u00e9m disso, voc\u00ea encontrar\u00e1 as propriedades deste tipo de matrizes e exerc\u00edcios resolvidos passo a passo.<\/p>\n<h2 class=\"wp-block-heading\"> O que \u00e9 uma matriz normal?<\/h2>\n<p> A defini\u00e7\u00e3o normal da matriz \u00e9: <\/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 align=\"LEFT\"> Uma <strong>matriz normal<\/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 ao produto da transposta conjugada por ela mesma.<\/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-196deb39b1de9764cb4013ded78fe671_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cdot A^*=A^*\\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"118\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p 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-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> \u00e9 a matriz transposta conjugada de<\/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> .<\/p>\n<\/div>\n<p> Por\u00e9m, se forem matrizes <strong>de n\u00fameros reais<\/strong> , a condi\u00e7\u00e3o anterior equivale a dizer que uma matriz comuta com a sua transposta, ou seja:<\/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-4f808080cda647c3e7cbf2cac2129539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cdot A^t=A^t\\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"114\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Porque, obviamente, a matriz transposta conjugada de uma matriz real \u00e9 simplesmente a matriz transposta (ou transposta).<\/p>\n<h2 class=\"wp-block-heading\"> Exemplos de matrizes normais<\/h2>\n<h3 class=\"wp-block-heading\"> Exemplo com n\u00fameros complexos<\/h3>\n<p> A seguinte matriz quadrada complexa de dimens\u00e3o 2\u00d72 \u00e9 normal: <\/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-matrice-normale-complexe-22152-1.webp\" alt=\"exemplo de matriz normal com n\u00fameros complexos de dimens\u00e3o 2x2\" class=\"wp-image-2041\" width=\"125\" height=\"65\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> A demonstra\u00e7\u00e3o de sua normalidade est\u00e1 anexada abaixo:<\/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-f44b98cec879a8332c462d2393fbfbba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^* = \\begin{pmatrix} i &amp; i \\\\[1.1ex] i &amp; -i \\end{pmatrix} \\cdot \\begin{pmatrix} -i &amp; -i \\\\[1.1ex] -i &amp; i \\end{pmatrix} =\\begin{pmatrix} 2 &amp; 0 \\\\[1.1ex] 0 &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"319\" 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-fddc406493ac1c81c86edf1ad6e58d0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*\\cdot A = \\begin{pmatrix} -i &amp; -i \\\\[1.1ex] -i &amp; i \\end{pmatrix}\\cdot \\begin{pmatrix} i &amp; i \\\\[1.1ex] i &amp; -i \\end{pmatrix}  = \\begin{pmatrix} 2 &amp; 0 \\\\[1.1ex] 0 &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"319\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo com n\u00fameros reais<\/h3>\n<p> A seguinte matriz quadrada com n\u00fameros reais de ordem 2 tamb\u00e9m \u00e9 normal: <\/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\/example-real-normal-matrix-22152-1.webp\" alt=\"exemplo de matriz normal com n\u00fameros reais de dimens\u00e3o 2x2\" class=\"wp-image-2042\" width=\"130\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Neste caso, por s\u00f3 possuir n\u00fameros reais, para provar que \u00e9 normal basta verificar que a matriz \u00e9 comut\u00e1vel com a sua transposta:<\/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-a320a8e300315c6a48bb8095266408ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B\\cdot B^t = \\begin{pmatrix} 2 &amp; -2 \\\\[1.1ex] 2 &amp; 2 \\end{pmatrix} \\cdot \\begin{pmatrix} 2 &amp; 2 \\\\[1.1ex] -2 &amp; 2 \\end{pmatrix} =\\begin{pmatrix} 8 &amp; 0 \\\\[1.1ex] 0 &amp; 8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"317\" 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-b6ad5bd62deeb5bcbf561a2ee6b29741_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B^t\\cdot B =\\begin{pmatrix} 2 &amp; 2 \\\\[1.1ex] -2 &amp; 2 \\end{pmatrix}\\cdot \\begin{pmatrix} 2 &amp; -2 \\\\[1.1ex] 2 &amp; 2 \\end{pmatrix} =\\begin{pmatrix} 8 &amp; 0 \\\\[1.1ex] 0 &amp; 8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"317\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Propriedades de matrizes normais<\/h2>\n<p> Matrizes normais t\u00eam as seguintes caracter\u00edsticas:<\/p>\n<ul>\n<li> Todas as matrizes normais s\u00e3o matrizes diagonaliz\u00e1veis.<\/li>\n<\/ul>\n<ul>\n<li> Cada <a href=\"https:\/\/mathority.org\/pt\/matriz-unitaria\/\">matriz unit\u00e1ria<\/a> tamb\u00e9m \u00e9 uma matriz normal.<\/li>\n<\/ul>\n<ul>\n<li> Da mesma forma, uma <a href=\"https:\/\/mathority.org\/pt\/matriz-hermitiana-ou-hermitiana\/\">matriz Hermitiana<\/a> \u00e9 uma matriz normal.<\/li>\n<\/ul>\n<ul>\n<li> Da mesma forma, uma matriz anti-hermitiana \u00e9 uma matriz normal.<\/li>\n<\/ul>\n<ul>\n<li> Se A \u00e9 uma matriz normal, os autovalores (ou autovalores) da matriz transposta conjugada A* s\u00e3o os autovalores conjugados de A.<\/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-a91ee46b5f8dda0d51ecb57474f5b816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}2i&amp;-1+i\\\\[1.1ex] 1+i&amp;i\\end{pmatrix} \\longrightarrow \\ \\lambda_{A,1} = 0 \\ ; \\ \\lambda_{A,2} = +3i\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"382\" 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-48c80a017a9afd8b4cf3923757f4e945_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*=\\begin{pmatrix}-2i&amp;1-i\\\\[1.1ex] -1-i&amp;-i\\end{pmatrix} \\longrightarrow \\ \\lambda_{A^*,1} = 0 \\ ; \\ \\lambda_{A^*,2} = -3i\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"403\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Nas matrizes normais, os autovetores (ou autovetores) associados aos diferentes autovalores s\u00e3o ortogonais.<\/li>\n<\/ul>\n<ul>\n<li> Se uma matriz \u00e9 composta apenas por n\u00fameros reais e \u00e9 <a href=\"https:\/\/mathority.org\/pt\/exemplos-e-propriedades-de-matrizes-simetricas\/\">sim\u00e9trica<\/a> , \u00e9 ao mesmo tempo uma matriz normal.<\/li>\n<\/ul>\n<ul>\n<li> Da mesma forma, uma <a href=\"https:\/\/mathority.org\/pt\/exemplos-e-propriedades-de-matrizes-antissimetricas\/\">matriz real antissim\u00e9trica<\/a> tamb\u00e9m \u00e9 uma matriz normal.<\/li>\n<\/ul>\n<ul>\n<li> Finalmente, qualquer matriz ortogonal formada por n\u00fameros reais tamb\u00e9m \u00e9 uma matriz normal.<\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"> Exerc\u00edcios resolvidos para matrizes normais<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Verifique se a seguinte matriz complexa de dimens\u00e3o 2 \u00d7 2 \u00e9 normal: <\/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-ff27d19373c5a4dc8e95472ec295c657_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}1&amp;2+3i\\\\[1.1ex] 2+3i&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"168\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para mostrar que a matriz \u00e9 normal devemos primeiro calcular 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-17c96c654ce5b978f90a905b973d5ae7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*=\\begin{pmatrix}1&amp;2-3i\\\\[1.1ex] 2-3i&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"176\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora fazemos a verifica\u00e7\u00e3o multiplicando a matriz A pela matriz A* nas duas dire\u00e7\u00f5es poss\u00edveis: <\/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-36212e1d12cf35ea5dd27bd91d77ee56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^* = \\begin{pmatrix}1&amp;2+3i\\\\[1.1ex] 2+3i&amp;1\\end{pmatrix}\\cdot \\begin{pmatrix}1&amp;2-3i\\\\[1.1ex] 2-3i&amp;1\\end{pmatrix} = \\begin{pmatrix}14&amp;4\\\\[1.1ex] 4&amp;14\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"453\" 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-3db0fc8fdc948037452b4c6275896686_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*\\cdot A =\\begin{pmatrix}1&amp;2-3i\\\\[1.1ex] 2-3i&amp;1\\end{pmatrix}\\cdot  \\begin{pmatrix}1&amp;2+3i\\\\[1.1ex] 2+3i&amp;1\\end{pmatrix} = \\begin{pmatrix}14&amp;4\\\\[1.1ex] 4&amp;14\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"453\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O resultado de ambas as multiplica\u00e7\u00f5es \u00e9 o mesmo, ent\u00e3o <strong>a matriz A \u00e9 normal.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 2<\/h3>\n<p> Mostre que a seguinte matriz real de tamanho 2 \u00d7 2 \u00e9 normal: <\/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-854e13859be417985691b5ed6d2a050f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}3&amp;5\\\\[1.1ex] -5&amp;3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"109\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Como neste caso se trata de um ambiente apenas com n\u00fameros reais, basta verificar que o produto matricial entre a matriz A e sua transposta d\u00e1 o mesmo resultado qualquer que seja o sentido da multiplica\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-b1b6314188f394b3053d3dac0613cf5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t = \\begin{pmatrix}3&amp;5\\\\[1.1ex] -5&amp;3\\end{pmatrix}\\cdot \\begin{pmatrix}3&amp;-5\\\\[1.1ex] 5&amp;3\\end{pmatrix} = \\begin{pmatrix}34&amp;0\\\\[1.1ex] 0&amp;34\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"332\" 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-e2b33f892cd29c0ee232b88eaa4946cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^t\\cdot A = \\begin{pmatrix}3&amp;-5\\\\[1.1ex] 5&amp;3\\end{pmatrix}\\cdot \\begin{pmatrix}3&amp;5\\\\[1.1ex] -5&amp;3\\end{pmatrix} = \\begin{pmatrix}34&amp;0\\\\[1.1ex] 0&amp;34\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"332\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O resultado de ambos os produtos \u00e9 o mesmo, ent\u00e3o <strong>a matriz A \u00e9 normal.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 3<\/h3>\n<p> Determine se a seguinte matriz de n\u00fameros complexos de ordem 2 \u00e9 normal: <\/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-00075db37b045e08349f7d5b3f679570_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}4i&amp;-1+i\\\\[1.1ex] 1-i&amp;4i\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"164\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para verificar se a matriz \u00e9 normal, devemos primeiro calcular 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-0b39733376eb2aef269012eb1d6c24be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*=\\begin{pmatrix}-4i&amp;1+i\\\\[1.1ex] -1-i&amp;-4i\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora verificamos se a matriz A e sua transposta conjugada s\u00e3o comut\u00e1veis: <\/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-c207cb9842dacbaf9bc59d4aaff00473_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^* = \\begin{pmatrix}4i&amp;-1+i\\\\[1.1ex] 1-i&amp;4i\\end{pmatrix}\\cdot \\begin{pmatrix}-4i&amp;1+i\\\\[1.1ex] -1-i&amp;-4i\\end{pmatrix} = \\begin{pmatrix}18&amp;8i\\\\[1.1ex] -8i&amp;18\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"456\" 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-bcf52f3da81fd7c56b090604c2b6f368_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*\\cdot A =\\begin{pmatrix}-4i&amp;1+i\\\\[1.1ex] -1-i&amp;-4i\\end{pmatrix}\\cdot  \\begin{pmatrix}4i&amp;-1+i\\\\[1.1ex] 1-i&amp;4i\\end{pmatrix} = \\begin{pmatrix}18&amp;8i\\\\[1.1ex] -8i&amp;18\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"456\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O resultado de ambas as multiplica\u00e7\u00f5es \u00e9 o mesmo, ent\u00e3o <strong>a matriz A \u00e9 normal.<\/strong> <\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 4<\/h3>\n<p> Verifique se a seguinte matriz real de dimens\u00e3o 3\u00d73 \u00e9 normal: <\/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-92ee07759c3e6e88af5a68479b5833ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} -1&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;1\\\\[1.1ex] 1&amp;0&amp;-1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"164\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Sendo a matriz inteiramente composta por elementos reais, basta verificar que o produto matricial entre a matriz A e sua transposta \u00e9 independente do sentido da multiplica\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-dc7ee02c75239b430c7fc2418f43e343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t = \\begin{pmatrix} -1&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;1\\\\[1.1ex] 1&amp;0&amp;-1\\end{pmatrix} \\cdot\\begin{pmatrix}-1&amp;0&amp;1\\\\[1.1ex] 1&amp;-1&amp;0\\\\[1.1ex] 0&amp;1&amp;-1\\end{pmatrix}=\\begin{pmatrix}2&amp;-1&amp;-1\\\\[1.1ex] -1&amp;2&amp;-1\\\\[1.1ex] -1&amp;-1&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"495\" 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-e661b877ee225983c797584e2b61d429_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^t\\cdot A =\\begin{pmatrix}-1&amp;0&amp;1\\\\[1.1ex] 1&amp;-1&amp;0\\\\[1.1ex] 0&amp;1&amp;-1\\end{pmatrix}\\cdot \\begin{pmatrix} -1&amp;1&amp;0\\\\[1.1ex] 0&amp;-1&amp;1\\\\[1.1ex] 1&amp;0&amp;-1\\end{pmatrix}=\\begin{pmatrix}2&amp;-1&amp;-1\\\\[1.1ex] -1&amp;2&amp;-1\\\\[1.1ex] -1&amp;-1&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"495\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O resultado de ambos os produtos \u00e9 o mesmo, ent\u00e3o <strong>a matriz A \u00e9 normal.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 5<\/h3>\n<p> Determine se a seguinte matriz complexa de ordem 3\u00d73 \u00e9 normal: <\/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-81ca0ac1da07c151a62dcfb06b4be877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}4&amp;3-2i &amp; 5i \\\\[1.1ex] 3+2i &amp; 0 &amp; -1-3i \\\\[1.1ex] -5i &amp; -1+3i &amp; 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"260\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, calculamos a transposta conjugada da 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-fd0a2dfe1b8bfe18020ab68c1eb3bda6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^*=\\begin{pmatrix}4&amp;3-2i &amp; 5i \\\\[1.1ex] 3+2i &amp; 0 &amp; -1-3i \\\\[1.1ex] -5i &amp; -1+3i &amp; 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"268\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora precisamos fazer as multiplica\u00e7\u00f5es de matrizes entre a matriz A e sua transposta conjugada em ambas as dire\u00e7\u00f5es poss\u00edveis. No entanto, a matriz transposta conjugada de A \u00e9 igual \u00e0 pr\u00f3pria matriz A, portanto \u00e9 uma matriz Hermitiana. E, portanto, <strong>das propriedades das matrizes normais segue-se que A \u00e9 uma matriz normal<\/strong> , porque toda matriz hermitiana \u00e9 uma matriz normal.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 uma matriz normal, bem como exemplos de matrizes normais. Al\u00e9m disso, voc\u00ea encontrar\u00e1 as propriedades deste tipo de matrizes e exerc\u00edcios resolvidos passo a passo. O que \u00e9 uma matriz normal? A defini\u00e7\u00e3o normal da matriz \u00e9: Uma matriz normal \u00e9 uma matriz complexa que multiplicada por sua &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/matriz-normal\/\"> <span class=\"screen-reader-text\">Matriz regular<\/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-314","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 regular - 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-normal\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriz regular - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 uma matriz normal, bem como exemplos de matrizes normais. 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