{"id":307,"date":"2023-07-06T12:44:55","date_gmt":"2023-07-06T12:44:55","guid":{"rendered":"https:\/\/mathority.org\/pt\/matriz-diagonal\/"},"modified":"2023-07-06T12:44:55","modified_gmt":"2023-07-06T12:44:55","slug":"matriz-diagonal","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/matriz-diagonal\/","title":{"rendered":"Matriz diagonal"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 uma matriz diagonal e exemplos de matrizes diagonais. Al\u00e9m disso, voc\u00ea descobrir\u00e1 como operar com este tipo de matrizes, como calcular facilmente seus determinantes e como invert\u00ea-los. Existem tamb\u00e9m propriedades e aplica\u00e7\u00f5es de matrizes diagonais. E, finalmente, h\u00e1 as explica\u00e7\u00f5es de uma matriz bidiagonal e de uma matriz tridiagonal.<\/p>\n<h2 class=\"wp-block-heading\"> O que \u00e9 uma matriz diagonal?<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Uma <strong>matriz diagonal<\/strong> \u00e9 uma matriz quadrada em que todos os elementos que n\u00e3o est\u00e3o na diagonal principal s\u00e3o zero (0). Os elementos da diagonal principal podem ou n\u00e3o ser zero.<\/p>\n<p> Assim que soubermos a defini\u00e7\u00e3o exata de uma matriz diagonal, veremos exemplos de matrizes diagonais:<\/p>\n<h2 class=\"wp-block-heading\"> Exemplos de matrizes diagonais<\/h2>\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 diagonal de dimens\u00e3o 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\/exemple-matrice-diagonale-22152-1.webp\" alt=\"Exemplo de matriz diagonal 2x2\" class=\"wp-image-1728\" width=\"73\" height=\"74\" 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 matriz diagonal de ordem 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\/exemple-matrice-diagonale-32153-1.webp\" alt=\"Exemplo de matriz diagonal 3x3\" class=\"wp-image-1729\" width=\"125\" height=\"114\" 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 matriz diagonal 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\/exemple-matrice-diagonale-42154-1.webp\" alt=\"Exemplo de matriz diagonal 4x4\" class=\"wp-image-1730\" width=\"161\" height=\"143\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Esses tipos de matrizes s\u00e3o geralmente escritos indicando os elementos da diagonal:<\/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-14216c3a6fd6e7bfd4c9d78ac2a4765c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"diag(2,5,1) = \\left. \\begin{pmatrix} 2 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 5 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"196\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Opera\u00e7\u00f5es com matrizes diagonais<\/h2>\n<p> Uma das raz\u00f5es pelas quais as matrizes diagonais s\u00e3o t\u00e3o importantes para a \u00e1lgebra linear \u00e9 a facilidade com que permitem realizar c\u00e1lculos. \u00c9 por isso que eles s\u00e3o t\u00e3o usados em matem\u00e1tica.<\/p>\n<h3 class=\"wp-block-heading\"> Adicionando e subtraindo matrizes diagonais<\/h3>\n<p> Adicionar (e subtrair) duas matrizes diagonais \u00e9 muito simples: basta adicionar (ou subtrair) os n\u00fameros nas diagonais.<\/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-6f7d2e19d548ee0d53465992ebac7fb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{diag}(a_1,... ,a_n) \\pm \\text{diag}(b_1 ,... , b_n) = \\text{diag}(a_1\\pm b_1,..., a_n\\pm b_n)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"454\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> 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-e659649fca7fe55f33c0f3452e8c46f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 5 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; -2 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 6 \\end{pmatrix} +\\begin{pmatrix} 1 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 3 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; -4 \\end{pmatrix} = \\begin{pmatrix} 6&amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 1 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"333\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Multiplica\u00e7\u00e3o de matriz diagonal<\/h3>\n<p> Para resolver uma multiplica\u00e7\u00e3o ou produto matricial de duas matrizes diagonais, basta multiplicar os elementos das diagonais.<\/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-88d1625220fe5fd9bda3767f15b59372_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{diag}(a_1,... ,a_n) \\cdot \\text{diag}(b_1 ,... , b_n) = \\text{diag}(a_1\\cdot b_1,..., a_n\\cdot b_n)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"427\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> 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-0bcb4a59778cc41eed67dce0bc384682_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; -4 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; -3 \\end{pmatrix} \\cdot\\begin{pmatrix} 5 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; -2 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 6 \\end{pmatrix} = \\begin{pmatrix} 5 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 8 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; -18 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"361\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Poder de matrizes diagonais<\/h3>\n<p> Para calcular a pot\u00eancia de uma matriz diagonal, precisamos elevar cada elemento da diagonal ao expoente:<\/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-fcfe6475c0d4ea75691ed4c9bdaa64cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\text{diag}(a_1,... ,a_n)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -5px;\"><\/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-b93b6a0717632e9bee22dcc5f5799f63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^k= \\text{diag}(a_1^k,... ,a_n^k)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"157\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> 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-4d27337283f4b6029bff166fb8e3458d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\left. \\begin{pmatrix} 3 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 2 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 4 \\end{pmatrix}\\right.^3= \\begin{pmatrix} 27 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 8 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 64 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"221\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h2 class=\"wp-block-heading\"> Determinante de uma matriz diagonal<\/h2>\n<p> O <strong>determinante de uma matriz diagonal<\/strong> \u00e9 o produto dos elementos da diagonal principal.<\/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-fcfe6475c0d4ea75691ed4c9bdaa64cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\text{diag}(a_1,... ,a_n)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -5px;\"><\/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-326faa61bf2e51b299c2b0274c7c0416_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A)= \\prod_{i =1}^n a_i\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"115\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p> Veja o seguinte exerc\u00edcio resolvido no qual encontramos o determinante de uma matriz diagonal simplesmente multiplicando os elementos de sua diagonal principal:<\/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-f34514c6e1559b8ebb296ee6c51a33d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 5 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 2 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 3 \\end{vmatrix} = 5 \\cdot 2 \\cdot 3 = 30\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"186\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Este teorema \u00e9 f\u00e1cil de provar: basta calcular o determinante de uma matriz diagonal por blocos (ou cofatores). Esta <strong>demonstra\u00e7\u00e3o<\/strong> \u00e9 detalhada abaixo usando uma matriz diagonal gen\u00e9rica:<\/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-7b8718172b4b70d1ccacb01ea7ed5dd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\begin{vmatrix} a &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; b &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; c \\end{vmatrix}&amp;  = a \\cdot \\begin{vmatrix} b &amp; 0 \\\\[1.1ex] 0 &amp; c \\end{vmatrix} - 0 \\cdot \\begin{vmatrix} 0 &amp; 0 \\\\[1.1ex] 0 &amp; c \\end{vmatrix} + 0 \\cdot \\begin{vmatrix} 0 &amp; b \\\\[1.1ex] 0 &amp; 0 \\end{vmatrix} \\\\[2ex] &amp; =a \\cdot (b\\cdot c) - 0 \\cdot 0 + 0 \\cdot 0 \\\\[2ex] &amp; = a \\cdot b \\cdot c \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"337\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Inverter uma matriz diagonal<\/h2>\n<p> Uma matriz diagonal <strong>\u00e9 invert\u00edvel se e somente se todos os elementos da diagonal principal forem diferentes de 0<\/strong> . Neste caso dizemos que a matriz diagonal \u00e9 uma matriz regular.<\/p>\n<p> Al\u00e9m disso, a inversa de uma matriz diagonal ser\u00e1 sempre outra matriz diagonal com as <strong>inversas<\/strong> da diagonal principal:<\/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-a91beaaca82477a0c882b42da4eb7481_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} 3 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 2 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 8 \\end{pmatrix}  \\ \\longrightarrow \\ A^{-1}=\\begin{pmatrix} \\frac{1}{3} &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; \\frac{1}{2} &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; \\frac{1}{8} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"324\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Da caracter\u00edstica anterior, podemos deduzir que o determinante da inversa de uma matriz diagonal \u00e9 o produto das inversas da diagonal principal: <\/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-0571390802f955fac935aeb9cf4ab92f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B= \\begin{pmatrix} 2 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 4 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"137\" 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-ae2b6aa1dd4d6405d30753e66e7f7486_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\left| B^{-1}\\right|=\\cfrac{1}{2} \\cdot \\cfrac{1}{4} \\cdot \\cfrac{1}{-1}=-\\cfrac{1}{8} = -0,125\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"266\" style=\"vertical-align: -12px;\"><\/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 de matrizes diagonais<\/h2>\n<ul>\n<li> Qualquer matriz diagonal tamb\u00e9m \u00e9 uma <a href=\"https:\/\/mathority.org\/pt\/exemplos-e-propriedades-de-matrizes-simetricas\/\">matriz sim\u00e9trica<\/a> .<\/li>\n<\/ul>\n<ul>\n<li> Uma matriz diagonal \u00e9 uma <a href=\"https:\/\/mathority.org\/pt\/matriz-triangular-superior-inferior\/\">matriz triangular superior e inferior<\/a> .<\/li>\n<\/ul>\n<ul>\n<li> A <span style=\"color:#1976d2;\"><strong>matriz identidade<\/strong><\/span> \u00e9 uma matriz diagonal:<\/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-4e4e9931fb7ae104414006cee93978a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\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=\"80\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Da mesma forma, a <span style=\"color:#1976d2;\"><strong>matriz zero<\/strong><\/span> tamb\u00e9m \u00e9 uma matriz diagonal, pois todos os seus elementos que n\u00e3o est\u00e3o na diagonal s\u00e3o zeros. Embora os n\u00fameros na diagonal sejam 0.<\/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-edb061dcbc869eba51ece12af43f796f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 0 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"80\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Os <span style=\"color:#1976d2;\"><strong>autovalores (ou autovalores)<\/strong><\/span> de uma matriz diagonal s\u00e3o os elementos de sua diagonal principal.<\/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-1dea3de2ae28d46194ead012bc001cf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 7 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 3 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 4 \\end{pmatrix} \\longrightarrow \\ \\lambda = 3 \\ ; \\ \\lambda = 4 \\ ; \\ \\lambda = 7\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"298\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Uma matriz quadrada \u00e9 diagonal se e somente se for <span style=\"color:#1976d2;\"><strong>triangular e normal<\/strong><\/span> .<\/li>\n<\/ul>\n<ul>\n<li> O <span style=\"color:#1976d2;\"><strong>adjunto<\/strong><\/span> de uma matriz diagonal \u00e9 outra matriz diagonal. <\/li>\n<\/ul>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Aplica\u00e7\u00f5es de matriz diagonal<\/h2>\n<p> Como vimos, resolver c\u00e1lculos com matrizes diagonais \u00e9 muito simples, pois muitos zeros est\u00e3o envolvidos nas opera\u00e7\u00f5es. Por esse motivo, s\u00e3o muito \u00fateis no campo da matem\u00e1tica e s\u00e3o amplamente utilizados.<\/p>\n<p> Por esse mesmo motivo, tantos estudos foram feitos sobre como <strong>diagonalizar uma matriz<\/strong> e, de fato, at\u00e9 foi desenvolvido um m\u00e9todo para diagonalizar matrizes (usando o polin\u00f4mio caracter\u00edstico).<\/p>\n<p> Portanto, matrizes diagonaliz\u00e1veis tamb\u00e9m s\u00e3o bastante relevantes. Como o teorema da decomposi\u00e7\u00e3o espectral, que estabelece as condi\u00e7\u00f5es para quando uma matriz pode ser diagonalizada e quando n\u00e3o o \u00e9.<\/p>\n<h2 class=\"wp-block-heading\"> matriz bidiagonal<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Uma <strong>matriz bidiagonal<\/strong> \u00e9 uma matriz quadrada em que todos os elementos que n\u00e3o est\u00e3o na diagonal principal ou na diagonal superior ou inferior s\u00e3o 0.<\/p>\n<p> Por exemplo: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-25\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9acdfc09d0167548ef3f6f5b58d9276_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 3 &amp; 2 &amp; 0 \\\\[1.1ex] 0 &amp; -5 &amp; 1 \\\\[1.1ex] 0 &amp; 0 &amp; 6 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"94\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <strong>matriz bidiagonal superior<\/strong> <\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b2b53f238add73431696006f4b05a2d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 1 &amp; 0 &amp; 0 \\\\[1.1ex] 6 &amp; 2 &amp; 0 \\\\[1.1ex] 0 &amp; 7 &amp; 4 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"80\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <strong>matriz bidiagonal inferior<\/strong><\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Quando a diagonal principal e a primeira superdiagonal est\u00e3o ocupadas, falamos de uma matriz bidiagonal superior. Por outro lado, quando a diagonal principal e a primeira subdiagonal est\u00e3o ocupadas, falamos de uma matriz bidiagonal inferior.<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h2 class=\"wp-block-heading\"> matriz tridiagonal<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Uma <strong>matriz tridiagonal<\/strong> \u00e9 uma matriz quadrada cujos \u00fanicos elementos diferentes de zero s\u00e3o os da diagonal principal e as diagonais adjacentes acima e abaixo.<\/p>\n<p> 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-9a8fbe0404c447268a89ff954e3b23d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 2 &amp; 3 &amp; 0 &amp; 0  \\\\[1.1ex] -4 &amp; 5 &amp; 9 &amp; 0 \\\\[1.1ex] 0 &amp; 1 &amp; 6 &amp; -2 \\\\[1.1ex] 0 &amp; 0 &amp; 8 &amp; 7 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"133\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Assim, todas as matrizes diagonais, bidiagonais e tridiagonais s\u00e3o exemplos de <strong>matrizes de banda<\/strong> . Porque uma matriz de banda \u00e9 aquela matriz que possui todos os seus elementos diferentes de zero em torno da diagonal principal.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 uma matriz diagonal e exemplos de matrizes diagonais. Al\u00e9m disso, voc\u00ea descobrir\u00e1 como operar com este tipo de matrizes, como calcular facilmente seus determinantes e como invert\u00ea-los. Existem tamb\u00e9m propriedades e aplica\u00e7\u00f5es de matrizes diagonais. E, finalmente, h\u00e1 as explica\u00e7\u00f5es de uma matriz bidiagonal e de uma matriz &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/matriz-diagonal\/\"> <span class=\"screen-reader-text\">Matriz diagonal<\/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-307","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 diagonal - 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-diagonal\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriz diagonal - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 uma matriz diagonal e exemplos de matrizes diagonais. 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