{"id":306,"date":"2023-07-06T13:14:29","date_gmt":"2023-07-06T13:14:29","guid":{"rendered":"https:\/\/mathority.org\/pt\/matriz-triangular-superior-inferior\/"},"modified":"2023-07-06T13:14:29","modified_gmt":"2023-07-06T13:14:29","slug":"matriz-triangular-superior-inferior","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/matriz-triangular-superior-inferior\/","title":{"rendered":"Matriz triangular superior e inferior"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 uma matriz triangular e os diferentes tipos de matrizes triangulares junto com exemplos. Al\u00e9m disso, voc\u00ea descobrir\u00e1 como calcular o determinante de uma matriz triangular e quais s\u00e3o as propriedades dessa matriz muito interessante. Por fim, explicamos tamb\u00e9m o que \u00e9 uma matriz de Hessenberg, pois \u00e9 uma matriz relacionada a matrizes triangulares.<\/p>\n<h2 class=\"wp-block-heading\"> O que \u00e9 uma matriz triangular?<\/h2>\n<p> Defini\u00e7\u00e3o de matriz triangular:<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Uma <strong>matriz triangular<\/strong> \u00e9 uma matriz quadrada em que todos os elementos acima ou abaixo da diagonal principal s\u00e3o zero (0).<\/p>\n<p> Matrizes triangulares s\u00e3o muito utilizadas em c\u00e1lculos de \u00e1lgebra linear, pois inverter uma matriz triangular, calcular seu determinante, ou mesmo resolver sistemas de equa\u00e7\u00f5es lineares com este tipo de matrizes \u00e9 muito mais f\u00e1cil do que com matrizes que possuem elementos diferentes de 0 em todas as posi\u00e7\u00f5es. .<\/p>\n<h2 class=\"wp-block-heading\"> matriz triangular superior<\/h2>\n<p> Uma <strong>matriz triangular superior<\/strong> \u00e9 uma matriz quadrada cujos elementos abaixo da diagonal principal s\u00e3o zero (0).<\/p>\n<p> Exemplo de matriz triangular superior: <\/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-triangulaire-superieure.webp\" alt=\"exemplo de matriz triangular superior\" class=\"wp-image-1648\" width=\"130\" height=\"114\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"> matriz triangular inferior<\/h2>\n<p> Uma <strong>matriz triangular inferior<\/strong> \u00e9 uma matriz quadrada que possui um zero (0) em cada elemento que est\u00e1 acima da diagonal principal.<\/p>\n<p> Exemplo de uma matriz triangular inferior: <\/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-triangulaire-inferieure.webp\" alt=\"exemplo de matriz triangular inferior\" class=\"wp-image-1649\" width=\"143\" height=\"113\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> \u00c0s vezes, essas matrizes tamb\u00e9m s\u00e3o chamadas pela letra U, para a matriz triangular superior, e pela letra L, para a matriz triangular inferior. Embora esta nomenclatura seja usada principalmente em ingl\u00eas, na verdade o U significa <em>matriz triangular superior<\/em> e o L significa <em>matriz triangular inferior<\/em> .<\/p>\n<h2 class=\"wp-block-heading\"> Exemplos de matrizes triangulares<\/h2>\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <span style=\"text-decoration: underline;\">Matriz triangular 2 \u00d7 2 dimens\u00f5es<\/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-triangulaire-superieure-22152-1.webp\" alt=\"Exemplo de matriz triangular superior 2x2\" class=\"wp-image-1658\" width=\"75\" height=\"72\" 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;\">Matriz triangular 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\/matrice-triangulaire-inferieur-32153-1.webp\" alt=\"Exemplo de matriz triangular inferior 3x3\" class=\"wp-image-1659\" width=\"131\" height=\"117\" 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;\">matriz triangular 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-triangulaire-superieure-42154-1.webp\" alt=\"Exemplo de matriz triangular superior 4x4\" class=\"wp-image-1660\" width=\"197\" height=\"144\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"> Determinante de uma matriz triangular<\/h2>\n<p> O <strong>determinante de uma matriz triangular<\/strong> , seja ela triangular superior ou inferior, \u00e9 o produto dos elementos da diagonal principal.<\/p>\n<p> D\u00ea uma olhada no seguinte exerc\u00edcio resolvido como basta calcular a multiplica\u00e7\u00e3o dos elementos da diagonal principal da matriz triangular para encontrar seu determinante:<\/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-7503e88c4eaabd74347a4f79461a3ebe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2 &amp; 5 &amp; -6 \\\\[1.1ex] 0 &amp; 4 &amp; 9 \\\\[1.1ex] 0 &amp; 0 &amp; 3 \\end{vmatrix} = 2 \\cdot 4 \\cdot 3 = \\bm{24}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"200\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Este teorema \u00e9 facilmente demonstrado: basta calcular o determinante de uma matriz triangular por blocos (ou cofatores). Esta <strong>demonstra\u00e7\u00e3o<\/strong> \u00e9 detalhada abaixo usando uma matriz triangular 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-91281c322af35f07cfbfd6fe61fc3c58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\begin{vmatrix} a &amp; b &amp; c \\\\[1.1ex] 0 &amp; d &amp; e \\\\[1.1ex] 0 &amp; 0 &amp; f \\end{vmatrix}&amp;  = a \\cdot \\begin{vmatrix} d &amp; e \\\\[1.1ex] 0 &amp; f \\end{vmatrix} - b \\cdot \\begin{vmatrix} 0 &amp;  e \\\\[1.1ex] 0 &amp;  f \\end{vmatrix} + c \\cdot \\begin{vmatrix} 0 &amp; d \\\\[1.1ex] 0 &amp; 0 \\end{vmatrix} \\\\[2ex] &amp; =a \\cdot (d\\cdot f) - b \\cdot 0 + c \\cdot 0 \\\\[2ex] &amp; = a \\cdot d \\cdot f \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"170\" width=\"341\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Por outro lado, sabemos que uma matriz \u00e9 invert\u00edvel se o seu determinante for diferente de 0. Assim, se nenhum elemento da diagonal principal for 0, a matriz triangular tamb\u00e9m ser\u00e1 invert\u00edvel e, consequentemente, ser\u00e1 regular. matriz.<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h2 class=\"wp-block-heading\"> Propriedades da matriz triangular<\/h2>\n<p> Agora vamos ver quais s\u00e3o as propriedades das matrizes triangulares:<\/p>\n<ul>\n<li> O <span style=\"color:#1976d2;\"><strong>produto de duas matrizes triangulares superiores<\/strong><\/span> \u00e9 igual a uma matriz triangular superior. E vice-versa: multiplicar duas matrizes triangulares inferiores d\u00e1 outra matriz triangular inferior.<\/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-dfd46e0ab8070d1c4c544d384fcf0f84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 3 &amp; 1 &amp; 4 \\\\[1.1ex] 0 &amp; -1 &amp; 2 \\\\[1.1ex] 0 &amp; 0 &amp; 5 \\end{pmatrix} \\cdot \\begin{pmatrix} 6 &amp; 2 &amp; 1 \\\\[1.1ex] 0 &amp; 3 &amp; 5 \\\\[1.1ex] 0 &amp; 0 &amp; 9 \\end{pmatrix} = \\begin{pmatrix}18&amp;9&amp;44\\\\[1.1ex] 0&amp;-3&amp;13\\\\[1.1ex]0&amp;0&amp;45\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"343\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> A <span style=\"color:#1976d2;\"><strong>transposta de uma matriz triangular superior<\/strong><\/span> \u00e9 uma matriz triangular inferior e vice-versa: a transposta de uma matriz triangular inferior \u00e9 uma matriz triangular superior.<\/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-ca1b4a07e3136aa75d1a8026e5e7c1ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{pmatrix} 1 &amp; 2 &amp; 6 &amp; 3 \\\\[1.1ex] 0 &amp; 9 &amp; 4 &amp; 1  \\\\[1.1ex] 0 &amp; 0 &amp; -2 &amp; 8 \\\\[1.1ex] 0 &amp; 0 &amp; 0 &amp; 7 \\end{pmatrix}\\right.^{\\bm{t}} =  \\begin{pmatrix} 1 &amp; 0 &amp; 0 &amp; 0 \\\\[1.1ex] 2 &amp; 9 &amp; 0 &amp; 0 \\\\[1.1ex] 6 &amp; 4 &amp; -2 &amp; 0 \\\\[1.1ex] 3 &amp; 1 &amp; 8 &amp; 7 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"279\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Uma <span style=\"color:#1976d2;\"><strong>matriz triangular \u00e9 invert\u00edvel<\/strong><\/span> se todos os seus elementos da diagonal principal forem diferentes de zero, ou seja, se forem diferentes de zero. Nesse caso, a inversa de uma matriz triangular superior (inferior) tamb\u00e9m \u00e9 uma matriz triangular superior (inferior).<\/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-adafaa535a161d29c9bcb8a31a572dc2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{pmatrix}1&amp;0&amp;0\\\\[1.1ex] -3&amp;2&amp;0\\\\[1.1ex] 2&amp;4&amp;3\\end{pmatrix} \\right.^{-1} =\\begin{pmatrix}1&amp;0&amp;0\\\\[1.1ex] \\frac{3}{2}&amp;\\frac{1}{2}&amp;0\\\\[1.1ex] -\\frac{8}{3}&amp;-\\frac{2}{3}&amp;\\frac{1}{3}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"261\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Al\u00e9m disso, a diagonal principal da matriz invertida sempre conter\u00e1 os inversos dos elementos da diagonal principal da matriz triangular original.<\/p>\n<ul>\n<li> Qualquer matriz diagonal \u00e9 uma matriz triangular superior e uma matriz triangular inferior, por exemplo:<\/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-497726e030cc2af2c07b16fdf3544024_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 3 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 8 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; -2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"94\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Portanto, uma <a href=\"https:\/\/mathority.org\/pt\/matriz-escalar\/\">matriz escalar<\/a> tamb\u00e9m \u00e9 uma matriz triangular superior e inferior. Por exemplo, a matriz identidade:<\/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> Obviamente, a <a href=\"https:\/\/mathority.org\/pt\/matriz-nula-zero\/\">matriz zero<\/a> tamb\u00e9m \u00e9 uma matriz triangular superior e inferior, pois os elementos acima e abaixo da diagonal principal s\u00e3o 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) de uma matriz triangular<\/strong><\/span> s\u00e3o os elementos da 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-272d0e156e1f27c20348b171c984e390_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 5 &amp; 0 &amp; 0 \\\\[1.1ex] 1 &amp; 3 &amp; 0 \\\\[1.1ex] 2 &amp; 6 &amp; -2 \\end{pmatrix} \\longrightarrow \\ \\lambda = -2 \\ ; \\ \\lambda = 3 \\ ; \\ \\lambda = 5\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"325\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Uma <span style=\"color:#1976d2;\"><strong>matriz triangular superior ou inferior \u00e9 sempre capaz de diagonalizar<\/strong><\/span> com base em autovetores (ou autovetores).<\/li>\n<\/ul>\n<ul>\n<li> <span style=\"color:#1976d2;\"><strong>Qualquer matriz pode ser fatorada no produto de uma matriz triangular inferior e uma matriz triangular superior<\/strong><\/span> . Ou seja, qualquer matriz pode ser transformada em uma multiplica\u00e7\u00e3o de matrizes triangulares. Al\u00e9m disso, se a matriz for invert\u00edvel, esta transforma\u00e7\u00e3o \u00e9 \u00fanica. Para fatorar uma matriz, o m\u00e9todo de decomposi\u00e7\u00e3o LU \u00e9 frequentemente usado. <\/li>\n<\/ul>\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\"> Triangularizar uma matriz<\/h2>\n<p> Existem v\u00e1rios teoremas sobre matrizes que podem ser triangularizadas alterando a base. Por\u00e9m, aqui veremos como triangular uma matriz aplicando <strong>transforma\u00e7\u00f5es elementares nas retas<\/strong> , como no m\u00e9todo de Gauss.<\/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-f66a4f370b37168439de204c1b0b401c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} 1 &amp; 2 &amp; 4 \\\\[1.1ex] 2 &amp; -3 &amp; 5 \\\\[1.1ex]1 &amp; -1 &amp; 6 \\end{pmatrix} \\begin{array}{c} \\\\[1.1ex] \\xrightarrow{f_2 -2f_1}\\\\[1.1ex] \\xrightarrow{f_3 -f_1} \\end{array}  \\begin{pmatrix} 1 &amp; 2 &amp; 4 \\\\[1.1ex] 0 &amp; -7 &amp; -3 \\\\[1.1ex] 0 &amp; -3 &amp; 2 \\end{pmatrix}\\begin{array}{c} \\\\[1.1ex]\\\\[1.1ex] \\xrightarrow{7f_3 -3f_2} \\end{array}  \\begin{pmatrix} 1 &amp; 2 &amp; 4 \\\\[1.1ex] 0 &amp; -7 &amp; -3 \\\\[1.1ex] 0 &amp; 0 &amp; 23 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"496\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E desta forma j\u00e1 triangularizamos a matriz original.<\/p>\n<p> Lembre-se que as transforma\u00e7\u00f5es elementares autorizadas entre linhas no m\u00e9todo gaussiano s\u00e3o:<\/p>\n<ul>\n<li> Substitua uma linha pela combina\u00e7\u00e3o linear de outras linhas.<\/li>\n<li> Multiplique ou divida todos os termos consecutivos por um n\u00famero diferente de 0.<\/li>\n<li> Edite linhas de pedido.<\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"> Matriz de Hessenberg<\/h2>\n<p> A defini\u00e7\u00e3o da matriz de Hessenberg \u00e9 a seguinte:<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> A <strong>matriz de Hessenberg<\/strong> \u00e9 uma matriz \u201cquase\u201d triangular, ou seja, todos os seus elementos s\u00e3o zero a partir da primeira subdiagonal (matriz de Hessenberg superior) ou da primeira superdiagonal (matriz de Hessenberg inferior).<\/p>\n<p> Tenho certeza de que \u00e9 melhor compreendido com um exemplo de matriz de Hessenberg superior e outro exemplo de matriz de Hessenberg inferior: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-28\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <strong>Matriz de Hessenberg superior<\/strong> <\/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-e76ad0fae8a28b5e5f31535683e63df5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{pmatrix} 3 &amp; 5 &amp; 1 &amp; 4 \\\\[1.1ex] 8 &amp; 2 &amp; 7 &amp; 1 \\\\[1.1ex] 0 &amp; 6 &amp; 3 &amp; 5 \\\\[1.1ex] 0 &amp; 0 &amp; 1 &amp; 9 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"105\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#1976d2\"> <strong>Matriz inferior de Hessenberg<\/strong><\/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-a9b13730483eaf930193baeb953d1d3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{pmatrix} 2 &amp; 4 &amp; 0 &amp; 0 \\\\[1.1ex] 1 &amp; 9 &amp; 6 &amp; 0 \\\\[1.1ex] 3 &amp; 5 &amp; 1 &amp; 2 \\\\[1.1ex] 8 &amp; 2 &amp; 3 &amp; 7 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"105\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> Uma matriz que \u00e9 uma matriz de Hessenberg superior e inferior \u00e9 uma <a href=\"https:\/\/mathority.org\/pt\/matriz-diagonal\/\">matriz tridiagonal<\/a> .<\/p>\n<p> Esta matriz tem o nome de Karl Hessenberg, um proeminente engenheiro e matem\u00e1tico alem\u00e3o do s\u00e9culo XX.<\/p>\n<p> Por fim, este tipo de matriz tem a particularidade de que se for multiplicada por uma matriz triangular, o resultado \u00e9 sempre uma matriz de Hessenberg.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 uma matriz triangular e os diferentes tipos de matrizes triangulares junto com exemplos. Al\u00e9m disso, voc\u00ea descobrir\u00e1 como calcular o determinante de uma matriz triangular e quais s\u00e3o as propriedades dessa matriz muito interessante. Por fim, explicamos tamb\u00e9m o que \u00e9 uma matriz de Hessenberg, pois \u00e9 uma &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/matriz-triangular-superior-inferior\/\"> <span class=\"screen-reader-text\">Matriz triangular superior e inferior<\/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-306","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 triangular superior e inferior - 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\/matriz-triangular-superior-inferior\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriz triangular superior e inferior - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 uma matriz triangular e os diferentes tipos de matrizes triangulares junto com exemplos. Al\u00e9m disso, voc\u00ea descobrir\u00e1 como calcular o determinante de uma matriz triangular e quais s\u00e3o as propriedades dessa matriz muito interessante. 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