{"id":332,"date":"2023-07-06T05:26:21","date_gmt":"2023-07-06T05:26:21","guid":{"rendered":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/"},"modified":"2023-07-06T05:26:21","modified_gmt":"2023-07-06T05:26:21","slug":"exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/","title":{"rendered":"Matriz ortogonal"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que s\u00e3o matrizes ortogonais e a rela\u00e7\u00e3o que elas t\u00eam com a inversa de uma matriz. Voc\u00ea tamb\u00e9m ver\u00e1 v\u00e1rios exemplos para entend\u00ea-lo perfeitamente. Al\u00e9m disso, ensinamos a f\u00f3rmula que verifica qualquer matriz ortogonal, com a qual voc\u00ea saber\u00e1 como encontr\u00e1-la rapidamente. E, finalmente, voc\u00ea encontrar\u00e1 as propriedades e aplica\u00e7\u00f5es dessas matrizes espec\u00edficas, bem como um t\u00edpico exerc\u00edcio de exame resolvido.<\/p>\n<h2 class=\"wp-block-heading\"> O que \u00e9 uma matriz ortogonal?<\/h2>\n<p> A defini\u00e7\u00e3o de matriz ortogonal \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 ortogonal<\/strong> \u00e9 uma matriz quadrada de n\u00fameros reais que multiplicada por sua transposta (ou transposta) \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-6ce7debd9ea0083703f398f280e534f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cdot A^t = A^t \\cdot A =I\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"147\" 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-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 uma matriz ortogonal e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-afd3cedfe0f405ed9f2d585b5ac1d8cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<p> representa sua matriz transposta.<\/p>\n<\/div>\n<p> Para que esta condi\u00e7\u00e3o seja atendida, as colunas e linhas de uma matriz ortogonal devem ser vetores unit\u00e1rios ortogonais, ou seja, devem formar uma base ortonormal. Por esta raz\u00e3o, alguns matem\u00e1ticos tamb\u00e9m as chamam <strong>de matrizes ortonormais<\/strong> .<\/p>\n<h2 class=\"wp-block-heading\"> Inversa de uma matriz ortogonal<\/h2>\n<p> Outra forma de explicar o conceito de matriz ortogonal \u00e9 atrav\u00e9s da matriz inversa, pois <strong>a matriz transposta (ou transposta) de uma matriz ortogonal \u00e9 igual \u00e0 sua inversa.<\/strong><\/p>\n<p> Para compreender completamente este teorema, \u00e9 importante que voc\u00ea saiba como <a href=\"https:\/\/mathority.org\/pt\/matriz-inversa\/\">inverter uma matriz<\/a> . Neste link voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o detalhada da inversa de uma matriz, todas as suas propriedades e ainda tem exerc\u00edcios resolvidos passo a passo para praticar.<\/p>\n<p> A matriz inversa de uma matriz ortogonal pode facilmente ser demonstrada como equivalente \u00e0 sua transposta usando a condi\u00e7\u00e3o da matriz ortogonal e a propriedade principal das matrizes inversas:<\/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-36f7666e4730a6311c088c7e8d7f0f38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{c} A \\cdot A^t =I \\\\[2ex] A \\cdot A^{-1} = I\\end{array} \\right\\} \\longrightarrow \\ A^t=A^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"231\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, uma matriz ortogonal ser\u00e1 sempre uma <a href=\"https:\/\/mathority.org\/pt\/quando-e-uma-matriz-regular-ou-invertivel-exemplos-e-propriedades\/\">matriz invert\u00edvel<\/a> , ou seja, ser\u00e1 uma matriz regular ou n\u00e3o degenerada.<\/p>\n<p> A seguir veremos v\u00e1rios exemplos de matrizes ortogonais para finalizar a compreens\u00e3o do conceito de tudo.<\/p>\n<h2 class=\"wp-block-heading\"> Exemplo de matriz ortogonal 2\u00d72<\/h2>\n<p> A seguinte matriz \u00e9 uma matriz ortogonal de dimens\u00e3o 2\u00d72: <\/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-orthogonale-de-dimension-22152-1.webp\" alt=\"matriz ortogonal de dimens\u00e3o 2x2\" class=\"wp-image-3302\" width=\"132\" height=\"70\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Podemos verificar que \u00e9 ortogonal calculando o produto pela 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-d157361ae2a13dbeabc4ba1aab7f8a94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"44\" 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-f7baa091c2fd963507b93e6bec5c386b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t= \\begin{pmatrix} 0 &amp; 1 \\\\[1.1ex] -1 &amp; 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"315\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Como o resultado d\u00e1 a matriz Id\u00eantica, verificamos que A \u00e9 uma matriz ortogonal.<\/p>\n<h2 class=\"wp-block-heading\"> Exemplo de matriz ortogonal 3\u00d73<\/h2>\n<p> A seguinte matriz \u00e9 uma matriz ortogonal de dimens\u00e3o 3\u00d73: <\/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-orthogonale-de-dimension-32153-1.webp\" alt=\"matriz ortogonal de dimens\u00e3o 3x3\" class=\"wp-image-3304\" width=\"193\" height=\"102\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Podemos mostrar que \u00e9 ortogonal multiplicando a matriz A pela 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-35687f56ff4ad5d1b19ea673b4ac85de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t = \\begin{pmatrix}0.8&amp;0.6&amp;0\\\\[1.1ex] -0.6&amp;0.8&amp;0\\\\[1.1ex] 0&amp;0&amp;1\\end{pmatrix}\\cdot \\begin{pmatrix}0.8&amp;-0.6&amp;0\\\\[1.1ex] 0.6&amp;0.8&amp;0\\\\[1.1ex] 0&amp;0&amp;1\\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=\"453\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Como a solu\u00e7\u00e3o \u00e9 a matriz unit\u00e1ria, mostramos que A \u00e9 uma matriz ortogonal.<\/p>\n<h2 class=\"wp-block-heading\"> F\u00f3rmula para encontrar uma matriz ortogonal 2&#215;2<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Veremos ent\u00e3o a prova de que todas as matrizes ortogonais de ordem 2 seguem o mesmo padr\u00e3o.<\/p>\n<p> Considere uma matriz gen\u00e9rica de tamanho 2\u00d72:<\/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-ac19d6ab63d390a9340cbce4014b1136_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} a &amp; b \\\\[1.1ex] c &amp; d \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"96\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Para que esta matriz seja ortogonal, a seguinte equa\u00e7\u00e3o matricial deve ser satisfeita:<\/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-05ba7bc31dc95f239c8ddb0ffdd72a81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t =I\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" 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-1e108701513ef6f2118e3b7d32657cd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} a &amp; b \\\\[1.1ex] c &amp; d \\end{pmatrix} \\cdot \\begin{pmatrix} a &amp; c \\\\[1.1ex] b &amp; d \\end{pmatrix} =\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Resolvendo a multiplica\u00e7\u00e3o de matrizes, obtemos as seguintes equa\u00e7\u00f5es:<\/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-d5435c614cb0da442fe04f65aec89637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} a^2+b^2 &amp; ac+bd \\\\[1.1ex] ac+bd &amp; c^2+d^2 \\end{pmatrix}=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"233\" 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-f8897132ecdbf389450e8c5fa1707226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{c}a^2+b^2=1 \\\\[2ex] ac+bd=0 \\\\[2ex] c^2+d^2=1 \\end{array} \\qquad \\begin{array}{l} (1) \\\\[2ex] (2) \\\\[2ex] (3) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"162\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Se voc\u00ea olhar de perto, essas igualdades se parecem muito com a <em>rela\u00e7\u00e3o trigonom\u00e9trica pitag\u00f3rica fundamental<\/em> :<\/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-cbd8ab83a790807844d1d30e63429337_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\sin ^2\\alpha+\\cos ^2\\alpha=1\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Consequentemente, os termos que satisfazem as equa\u00e7\u00f5es (1) e (3) obtidas s\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-9abeb023c5050d8d7f6fbab8c52227ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{l} a = \\cos \\theta \\qquad \\qquad \\qquad c = \\sin\\phi \\\\[2ex] b = \\sin \\theta \\qquad \\qquad \\qquad d = \\cos \\phi\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"242\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Al\u00e9m disso, substituindo os valores na segunda equa\u00e7\u00e3o, obtemos a rela\u00e7\u00e3o entre os dois \u00e2ngulos: <\/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-9b210cbf7eb8602c723c54204fc5ad8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle ac+bd=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"88\" 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-8c5edfeb3556ee37b43da4afaeb0c3f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos\\theta\\sin\\phi+\\sin\\theta\\cos\\phi=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"202\" style=\"vertical-align: -4px;\"><\/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-b2b1e1085946911044e2758ca2783eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\tan\\phi=-\\tan\\theta\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Ou seja, uma das duas condi\u00e7\u00f5es a seguir deve ser 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-0eb6746afdcb971294de82ecebad37b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{si} \\quad c=\\sin\\phi=-\\sin\\theta \\quad \\longrightarrow \\quad  d=\\cos\\phi=\\cos\\theta\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"373\" style=\"vertical-align: -4px;\"><\/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-6face2f33e95163135f12204424969f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{si} \\quad d=\\cos \\phi=-\\cos \\theta \\quad \\longrightarrow \\quad c=\\sin\\phi=\\sin\\theta\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"373\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Portanto, em conclus\u00e3o, as matrizes ortogonais devem ter a estrutura de uma das duas matrizes a seguir: <\/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\/formule-de-la-matrice-orthogonale-de-dimension-22152-1.webp\" alt=\"f\u00f3rmula para a matriz ortogonal de dimens\u00e3o 2x2\" class=\"wp-image-3267\" width=\"623\" height=\"143\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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-356a08e839ab6974a16448e16e56745d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\theta\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 um n\u00famero real.<\/p>\n<p> Com efeito, se a t\u00edtulo de exemplo concedermos o valor<\/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-d3f94c7fef174aa94efe99c9aa192cab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\theta=\\frac{\\pi}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"45\" style=\"vertical-align: -12px;\"><\/p>\n<p> e pegarmos a primeira estrutura, obteremos a matriz que verificamos ser ortogonal na se\u00e7\u00e3o \u201cExemplo de matriz ortogonal 2\u00d72\u201d: <\/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-1a331cab64745933f7c8a5009c799be6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle M_1 \\left(\\theta =\\frac{\\pi}{2}\\right)=\\begin{pmatrix} \\cos \\cfrac{\\pi}{2} &amp;\\sin \\cfrac{\\pi}{2} \\\\[4ex] -\\sin \\cfrac{\\pi}{2} &amp; \\cos \\cfrac{\\pi}{2} \\end{pmatrix}=\\begin{pmatrix} \\vphantom{\\frac{\\pi}{2}}0 &amp;1 \\\\[2ex]\\vphantom{\\frac{\\pi}{2}} -1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"366\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Propriedades da Matriz Ortogonal<\/h2>\n<p> As caracter\u00edsticas deste tipo de matriz s\u00e3o:<\/p>\n<ul>\n<li> Uma matriz ortogonal nunca pode ser uma <a href=\"https:\/\/mathority.org\/pt\/matriz-singular-ou-degenerada\/\">matriz singular<\/a> , porque sempre pode ser invertida. Nesse sentido, a inversa de uma matriz ortogonal \u00e9 outra matriz ortogonal.<\/li>\n<\/ul>\n<ul>\n<li> Qualquer matriz ortogonal pode ser diagonalizada. Dizemos ent\u00e3o que matrizes ortogonais s\u00e3o <em>ortogonalmente diagonaliz\u00e1veis.<\/em><\/li>\n<\/ul>\n<ul>\n<li> Todos os autovalores ou autovalores de uma matriz ortogonal t\u00eam m\u00f3dulo igual a 1.<\/li>\n<\/ul>\n<ul>\n<li> Qualquer matriz ortogonal composta apenas por n\u00fameros reais tamb\u00e9m \u00e9 uma matriz normal.<\/li>\n<\/ul>\n<ul>\n<li> O an\u00e1logo da matriz ortogonal em um ambiente com n\u00fameros complexos \u00e9 a matriz unit\u00e1ria.<\/li>\n<\/ul>\n<ul>\n<li> Obviamente, a matriz identidade \u00e9 uma matriz ortogonal.<\/li>\n<\/ul>\n<ul>\n<li> O conjunto de matrizes ortogonais de dimens\u00e3o n \u00d7 n bem como a opera\u00e7\u00e3o do produto matricial formam um grupo denominado grupo ortogonal. Ou seja, o produto de duas matrizes ortogonais \u00e9 igual a outra matriz ortogonal.<\/li>\n<\/ul>\n<ul>\n<li> Al\u00e9m disso, o resultado da multiplica\u00e7\u00e3o de uma matriz ortogonal pela sua transposta pode ser expresso pelo delta de Kronecker:<\/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-7d0922008f857f33f46de7551a8ff7cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A\\cdot A^{t})_{ij} = \\delta_{ij}=\\begin{cases}1 &amp; \\mbox{si }i = j, \\\\[2ex] 0 &amp; \\mbox{si }i \\ne j\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"238\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Finalmente, o determinante de uma matriz ortogonal \u00e9 sempre +1 ou -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-12d5717a2cb94708642478117c7c309d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A)=\\pm 1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Exerc\u00edcio resolvido de matrizes ortogonais<\/h2>\n<p> Resolveremos ent\u00e3o um exerc\u00edcio sobre matrizes ortogonais.<\/p>\n<ul>\n<li> Dada a seguinte matriz quadrada de ordem 3, encontre os valores de\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" 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-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> para torn\u00e1-lo ortogonal:<\/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-892ca58ec5cd36060396cb566902d65d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\frac{1}{3}\\begin{pmatrix}a&amp;a&amp;1\\\\[1.1ex] b&amp;1&amp;b\\\\[1.1ex] 1&amp;a&amp;a\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"140\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Para que a ortogonalidade da matriz seja cumprida, o produto da matriz pela sua transposta deve ser igual \u00e0 matriz Identidade. ENT\u00c3O:<\/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-301d774ec2d0663c858c91e548000749_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t = I\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" 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-c2dc9ef8c514302f183ca66626cabc1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{3}\\begin{pmatrix}a&amp;a&amp;1\\\\[1.1ex] b&amp;1&amp;b\\\\[1.1ex] 1&amp;a&amp;a\\end{pmatrix} \\cdot \\frac{1}{3}\\begin{pmatrix}a&amp;b&amp;1\\\\[1.1ex] a&amp;1&amp;a\\\\[1.1ex] 1&amp;b&amp;a\\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=\"334\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Multiplicamos 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-18a21a22f3cc9747c310d271c3fe4c5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{9}\\begin{pmatrix}2a^2+1&amp;ab+a+b&amp;2a+a^2\\\\[1.5ex] ab+a+b&amp;2b^2+1&amp;b+a+ab\\\\[1.5ex] 2a+a^2&amp;b+a+ab&amp;1+2a^2\\end{pmatrix} =\\begin{pmatrix}1&amp;0&amp;0\\\\[1.5ex] 0&amp;1&amp;0\\\\[1.5ex] 0&amp;0&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"418\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Agora podemos obter uma equa\u00e7\u00e3o do canto superior esquerdo das matrizes, porque os elementos nessa posi\u00e7\u00e3o devem corresponder. Ainda: <\/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-3a4d0f699410c3a7de5d6af181073f8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{9}(2a^2+1) = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"113\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<p> Resolvemos a equa\u00e7\u00e3o e eliminamos a inc\u00f3gnita: <\/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-c6700a19d58afe1d84668664734ef725_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2a^2+1 = 9\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" 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-a2007d061c6ecfda4deef22882bfce17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2a^2 = 8\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" 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-5fd565f4a4ab39908601493ada1575cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a^2 = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" 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-2f617bc4be3760ed1e13564672a4b6ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{a = \\pm 2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> No entanto, existem equa\u00e7\u00f5es que n\u00e3o se sustentam com a solu\u00e7\u00e3o positiva, por exemplo, aquela no canto superior direito. Portanto <strong>, apenas a solu\u00e7\u00e3o negativa \u00e9 poss\u00edvel<\/strong> .<\/p>\n<p> Por outro lado, para calcular a vari\u00e1vel<\/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> podemos combinar, por exemplo, os termos colocados na segunda linha da primeira coluna:<\/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-16d30b554d24bd4b8a00b156ed1503d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{9}(ab+a+b) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"134\" style=\"vertical-align: -12px;\"><\/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-fe062334bff68f2ba70f0f025d2a2d9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle ab+a+b = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"110\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Ao substituir o valor 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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> na 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-e8e6825541c5d7e2a12df99d9dc7b3b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2b-2+b = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"122\" 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-a13af6b471ccdb2e6804fc02b87abc4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -b =2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" 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-27e8de15a4a0f239878f4cc4f5b8db24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{b =-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"53\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Em suma, a \u00fanica solu\u00e7\u00e3o poss\u00edvel \u00e9:<\/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-70dc1f4331f63bff4d12f4bad8ef34e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{a=b =-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"86\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o a matriz ortogonal que corresponde a esses valores \u00e9:<\/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-cb7e7a27658da85f7b0d16b17f1f0815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\frac{1}{3}\\begin{pmatrix}-2&amp;-2&amp;1\\\\[1.1ex] -2&amp;1&amp;-2\\\\[1.1ex] 1&amp;-2&amp;-2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"179\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Aplica\u00e7\u00f5es de matrizes ortogonais<\/h2>\n<p> Embora possa n\u00e3o parecer porque geralmente t\u00eam uma forma muito simples, as matrizes ortogonais s\u00e3o muito importantes na matem\u00e1tica, especialmente no campo da \u00e1lgebra linear.<\/p>\n<p> Na geometria, as matrizes ortogonais representam transforma\u00e7\u00f5es isom\u00e9tricas (que n\u00e3o alteram dist\u00e2ncias e \u00e2ngulos) em espa\u00e7os vetoriais reais, por isso s\u00e3o chamadas de transforma\u00e7\u00f5es ortogonais. Al\u00e9m disso, estas transforma\u00e7\u00f5es s\u00e3o isomorfismos internos do espa\u00e7o vetorial considerado. Estas transforma\u00e7\u00f5es podem ser <strong>rota\u00e7\u00f5es<\/strong> , <strong>reflex\u00f5es especulares<\/strong> ou <strong>invers\u00f5es<\/strong> .<\/p>\n<p> Por fim, esse tipo de matriz tamb\u00e9m \u00e9 utilizado na f\u00edsica, pois permite estudar o movimento de corpos r\u00edgidos. E s\u00e3o at\u00e9 usados na formula\u00e7\u00e3o de certas teorias de campo.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que s\u00e3o matrizes ortogonais e a rela\u00e7\u00e3o que elas t\u00eam com a inversa de uma matriz. Voc\u00ea tamb\u00e9m ver\u00e1 v\u00e1rios exemplos para entend\u00ea-lo perfeitamente. Al\u00e9m disso, ensinamos a f\u00f3rmula que verifica qualquer matriz ortogonal, com a qual voc\u00ea saber\u00e1 como encontr\u00e1-la rapidamente. E, finalmente, voc\u00ea encontrar\u00e1 as propriedades e aplica\u00e7\u00f5es &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/\"> <span class=\"screen-reader-text\">Matriz ortogonal<\/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":[36],"tags":[],"class_list":["post-332","post","type-post","status-publish","format-standard","hentry","category-matriz-inversa"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matriz ortogonal - 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\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriz ortogonal - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea ver\u00e1 o que s\u00e3o matrizes ortogonais e a rela\u00e7\u00e3o que elas t\u00eam com a inversa de uma matriz. Voc\u00ea tamb\u00e9m ver\u00e1 v\u00e1rios exemplos para entend\u00ea-lo perfeitamente. Al\u00e9m disso, ensinamos a f\u00f3rmula que verifica qualquer matriz ortogonal, com a qual voc\u00ea saber\u00e1 como encontr\u00e1-la rapidamente. E, finalmente, voc\u00ea encontrar\u00e1 as propriedades e aplica\u00e7\u00f5es &hellip; Matriz ortogonal Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T05:26:21+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6ce7debd9ea0083703f398f280e534f3_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=\"5 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-ortogonais-propriedades-2x2-3x3\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Matriz ortogonal\",\"datePublished\":\"2023-07-06T05:26:21+00:00\",\"dateModified\":\"2023-07-06T05:26:21+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/\"},\"wordCount\":1028,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Matriz inversa\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/\",\"url\":\"https:\/\/mathority.org\/pt\/exemplos-de-matrizes-ortogonais-propriedades-2x2-3x3\/\",\"name\":\"Matriz ortogonal - 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