{"id":723,"date":"2025-12-15T08:25:43","date_gmt":"2025-12-15T07:25:43","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=723"},"modified":"2025-12-14T13:22:17","modified_gmt":"2025-12-14T12:22:17","slug":"alg-diagonalizacion-de-una-matriz","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=723","title":{"rendered":"ALG: Diagonalizaci\u00f3n de una matriz"},"content":{"rendered":"<p>Dado \\(\\mathbf {A} \\in M_{n\\times n}(\\mathbb {K} )\\), una matriz cuadrada con valores sobre un cuerpo \\(\\mathbb {K}\\), decimos que \\(\\mathbf{A}\\) es diagonalizable si, y s\u00f3lo si, \\(\\mathbf{A}\\) se puede descomponer de la forma: \\[\\mathbf{A}=\\mathbf{P}\\mathbf{D}\\mathbf{P}^{-1},\\] donde \\(\\mathbf{D}\\) es una matriz diagonal.<\/p>\n<p>El proceso de diagonalizaci\u00f3n de una matriz necesita conocer los autovalores y autovectores de la matriz. Sea, por tanto, \\(\\mathbf{A}\\) una matriz cuadrada  de orden \\(n\\), y sean \\(\\lambda_i\\) los autovalores de dicha matriz. Entonces<\/p>\n<blockquote>\n<p>La matriz \\(\\mathbf{A}\\) es diagonalizable si, y s\u00f3lo si, se cumple: <\/p>\n<ol>\n<li>el n\u00famero de soluciones de la ecuaci\u00f3n caracter\u00edstica es igual a \\(n\\); <\/li>\n<li>para todo autovalor \\(\\lambda_i\\), la dimensi\u00f3n del subespacio \\(\\mathcal{C}_{\\lambda_i}\\) coincide con la multiplicidad del autovalor \\(\\lambda_i\\) como soluci\u00f3n de la ecuaci\u00f3n caracter\u00edstica de \\(\\mathbf{A}\\); es decir, \\(m_g(\\lambda_i)= m_a(\\lambda)\\)<\/li>\n<\/ol>\n<\/blockquote>\n<blockquote><p><strong>Ejercicio:<\/strong> Es diagonalizable la matriz \\[\\begin{bmatrix}1 &#038; 0 &#038; 0\\\\<br \/>\n2 &#038; -1 &#038; 0\\\\<br \/>\n1 &#038; -1 &#038; 1\\end{bmatrix}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4r5() {\n  var htmlShow4r5 = document.getElementById(\"html-show4r5\");\n  if (htmlShow4r5.style.display === \"none\") {\n    htmlShow4r5.style.display = \"block\";\n  } else {\n    htmlShow4r5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4r5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4r5\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">Veamos quienes son los autovalores:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">factor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0\\\\2 &amp; -1 &amp; 0\\\\1 &amp; -1 &amp; 1\\end{bmatrix}\\]<\/p>\n<p>\\[-{{\\left( x-1\\right) }^{2}} \\left( x+1\\right) =0\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos los autovalores 1 y -1. El autovalor 1 es de multiplicidad algebraica dos, por tanto ser\u00e1 diagonalizable si su multiplicidad geom\u00e9trica es dos:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">rank<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[1\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Por tanto, la dimensi\u00f3n del subespacio propio es 2 (3-1) y es diagonalizable.<\/div>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Para qu\u00e9 valores de \\(a\\in\\mathbb{R}\\) no es diagonalizable la matriz \\[\\begin{bmatrix}4 &#038; 2 &#038; 0\\\\<br \/>\n3 &#038; 3 &#038; 0\\\\ 0 &#038; 0 &#038; a\\end{bmatrix}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4r15() {\n  var htmlShow4r15 = document.getElementById(\"html-show4r15\");\n  if (htmlShow4r15.style.display === \"none\") {\n    htmlShow4r15.style.display = \"block\";\n  } else {\n    htmlShow4r15.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4r15()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4r15\" style=\"display: none;\">\nCalculemos el polinomio caracter\u00edstico:<br \/>\n\\[p_\\lambda (A)=\\begin{vmatrix}4-\\lambda &#038; 2 &#038; 0\\\\<br \/>\n3 &#038; 3-\\lambda &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; a-\\lambda\\end{vmatrix}=(a-\\lambda)\\begin{vmatrix}4-\\lambda &#038; 2 \\\\<br \/>\n3 &#038; 3-\\lambda\\end{vmatrix}\\]<br \/>\nComo \\[\\begin{vmatrix}4-\\lambda &#038; 2 \\\\<br \/>\n3 &#038; 3-\\lambda\\end{vmatrix}\\overset{f_2-f_1}{=}\\begin{vmatrix}4-\\lambda &#038; 2 \\\\<br \/>\n-(1-\\lambda) &#038; 1-\\lambda\\end{vmatrix}=(1-\\lambda)\\begin{vmatrix}4-\\lambda &#038; 2 \\\\<br \/>\n-1 &#038; 1\\end{vmatrix}=(1-\\lambda)(6-\\lambda),\\]<br \/>\nresulta \\[p_\\lambda (A)=(a-\\lambda)(1-\\lambda)(6-\\lambda).\\]<br \/>\nSi \\(a\\neq 6\\) y \\(a\\neq 1\\) siempre ser\u00e1 diagonalizable. Pero si \\(a=6\\) o \\(a=1\\), la multiplicidad algebraica de ellos ser\u00e1 2; por tanto, su multiplicidad geom\u00e9trica tambi\u00e9n deber\u00e1 ser 2. Veamos que ocurre.<\/p>\n<p>Supongamos que \\(a=6\\), entonces se cumple que \\[\\mathbf{dim}\\ C_6=\\mathbf{dim}\\ \\mathbb{R}^3-\\mathbf{rank}(A-6\\mathbf{I}_3)\\]<br \/>\nY \\[\\mathbf{rank}\\left(\\begin{bmatrix}4 &#038; 2 &#038; 0\\\\<br \/>\n3 &#038; 3 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 6\\end{bmatrix}-6\\mathbf{I}_3\\right)=\\mathbf{rank}\\begin{bmatrix}-2 &#038; 2 &#038; 0\\\\<br \/>\n3 &#038; -3 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0\\end{bmatrix}=1.\\]<br \/>\nPor tanto, \\(\\mathbf{dim}\\ C_6=2\\) y la matriz es diagonalizable. <\/p>\n<p>Si \\(a=1\\):<br \/>\n\\[\\mathbf{rank}\\left(\\begin{bmatrix}4 &#038; 2 &#038; 0\\\\<br \/>\n3 &#038; 3 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 1\\end{bmatrix}-1\\mathbf{I}_3\\right)=\\mathbf{rank}\\begin{bmatrix}3 &#038; 2 &#038; 0\\\\3 &#038; 2 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0\\end{bmatrix}=1.\\]<\/p>\n<p>Luego para todo \\(a\\in\\mathbb{R}\\) la matriz es diagonizable.\n<\/p><\/div>\n<hr \/>\n<p>As\u00ed pues si  \\(\\mathbf{A}\\) es diagonalizable, ser\u00e1  \\(\\mathbf{D}\\) una matriz cuya diagonal principal est\u00e1 formada por los autovalores de \\(\\mathbf{A}\\) pareciendo cada uno tantas veces como indique su multiplicidad algebraica, y \\(\\mathbf{P}\\) es la matriz cuyas columnas son los autovectores; es decir, los vectores que constituyen una base del subespacio propio asociado a cada autovalor siguiendo el orden establecido en \\(\\mathbf{D}.\\)<\/p>\n<blockquote><p><strong>Ejercicio:<\/strong> Determinar la matriz que diagonaliza la matriz \\(\\begin{bmatrix}1&#038;0\\\\6&#038;-1\\end{bmatrix}\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4r() {\n  var htmlShow4r = document.getElementById(\"html-show4r\");\n  if (htmlShow4r.style.display === \"none\") {\n    htmlShow4r.style.display = \"block\";\n  } else {\n    htmlShow4r.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4r()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4r\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">factor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0\\\\6 &amp; -1\\end{bmatrix}\\]<\/p>\n<p>\\[\\left( x-1\\right) \\, \\left( x+1\\right) =0\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos los autovalores 1 y -1. Veamos cuales son los vectores propios asociados a su subespacio propio.<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%rnum_list<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}0\\\\6 x-2 y\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (1)\\]<\/p>\n<p>\\[\\left[ x=\\frac{{\\mathrm{\\% r1}}}{3},y={\\mathrm{\\% r1}}\\right] \\]<\/p>\n<p>\\[\\left[ \\frac{1}{3},1\\right] \\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%rnum_list<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}2 x\\\\6 x\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (2)\\]<\/p>\n<p>\\[\\left[ x=0,y={\\mathrm{\\% r2}}\\right] \\]<\/p>\n<p>\\[\\left[ 0,1\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">La matriz P ser\u00e1<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i11) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}0&amp;\\frac{1}{3} \\\\1 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i12) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">invert<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">P<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}-1 &amp; 0\\\\0 &amp; 1\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Sean \\(P\\) y \\(D\\), diagonal, talque la matriz \\[\\begin{bmatrix}2&#038;1\\\\1&#038;2\\end{bmatrix}=P.D.P^{-1}.\\] \u00bfCu\u00e1l es el determinante de \\(P.D\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2g5() {\n  var htmlShow2g5 = document.getElementById(\"html-show2g5\");\n  if (htmlShow2g5.style.display === \"none\") {\n    htmlShow2g5.style.display = \"block\";\n  } else {\n    htmlShow2g5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2g5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2g5\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">Veamos quienes son los autovalores:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">factor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 2\\\\2 &amp; 1\\end{bmatrix}\\]<\/p>\n<p>\\[\\left( x-3\\right) \\, \\left( x+1\\right) =0\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos los autovalores 3 y -1. Veamos cuales son los vectores propios asociados a su subespacio propio.<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%rnum_list<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}2 y-2 x\\\\2 x-2 y\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (2)\\]<\/p>\n<p>\\[\\left[ x={\\mathrm{\\% r1}},y={\\mathrm{\\% r1}}\\right] \\]<\/p>\n<p>\\[\\left[ 1,1\\right] \\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%rnum_list<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}2 y+2 x\\\\2 y+2 x\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (2)\\]<\/p>\n<p>\\[\\left[ x=-{\\mathrm{\\% r2}},y={\\mathrm{\\% r2}}\\right] \\]<\/p>\n<p>\\[\\left[ -1,1\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ya podemos construir la matriz P y calcular el determinante del producto P.D:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i13) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">D<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">D<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-6\\]<\/p>\n<\/div>\n<hr \/>\n<h3>Procedimiento para construir \\(P\\) y \\(D\\) en una diagonalizaci\u00f3n<\/h3>\n<p>Una vez que sabemos que la matriz \\(A\\in\\mathcal{M}_n(\\mathbb{R})\\) es diagonalizable, ordenamos los valores propios de menor a mayor, \\(\\lambda_1\\leq\\lambda_2\\leq\\ldots\\leq\\lambda_n\\). La matriz \\(D\\) ser\u00e1<br \/>\n\\[D=\\begin{bmatrix}<br \/>\n\\lambda _1 &#038; 0 &#038; \\cdots &#038; 0 \\\\<br \/>\n0 &#038; \\lambda _2 &#038; \\cdots &#038; 0 &#038;  \\\\<br \/>\n\\vdots  &#038; \\vdots  &#038; \\cdots  &#038; \\vdots  \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; \\lambda _n \\\\<br \/>\n\\end{bmatrix}.\\]<br \/>\nPara construir la matriz \\(P\\), a\u00f1adiremos como columnas los vectores propios normalizados asociados a los valores propios en orden.<\/p>\n<blockquote><p><strong>Ejercicio:<\/strong> Sean \\(P\\) la matriz de vectores propios normalizados de \\[\\begin{bmatrix}2 &#038; -1 &#038; 0\\\\<br \/>\n-4 &#038; 5 &#038; 0\\\\ 0 &#038; 0 &#038; 2\\end{bmatrix}.\\] Cu\u00e1l es la norma de la proyecci\u00f3n de \\(\\begin{bmatrix}3 &#038; -1 &#038; 2\\\\<br \/>\n-4 &#038; 5 &#038; 1\\\\ 1 &#038; 2 &#038; 2\\end{bmatrix}\\) sobre \\(P\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv23g5() {\n  var htmlShow23g5 = document.getElementById(\"html-show23g5\");\n  if (htmlShow23g5.style.display === \"none\") {\n    htmlShow23g5.style.display = \"block\";\n  } else {\n    htmlShow23g5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv23g5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show23g5\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">factor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-\\left( x-6\\right) \\, \\left( x-2\\right) \\, \\left( x-1\\right) \\]<\/p>\n<p>\\[\\left[ x=1,x=2,x=6\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos tres autovalores reales y distintos, luego la matriz es diagonalizable. Veamos su diagonalizaci\u00f3n.<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">sort<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">aut<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">for<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span><span class=\"code_function\">thru<\/span><span class=\"code_number\">3<\/span><span class=\"code_function\">do<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">a<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">l<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">a<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">linsolve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">%rnum_list<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}\\frac{1}{\\sqrt{2}} &amp; 0 &amp; \\frac{1}{\\sqrt{17}}\\\\\\frac{1}{\\sqrt{2}} &amp; 0 &amp; -\\frac{4}{\\sqrt{17}}\\\\0 &amp; 1 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">fpprintprec<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i13) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">B<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">mat_trace<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">B<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">P<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">mat_trace<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">P<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">P<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">mat_trace<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">float<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[0.4663\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Propiedades<\/strong><\/p>\n<p>Toda matriz sim\u00e9trica de coeficientes reales es diagonalizable y sus autovalores son reales.<\/p>\n<p>Dadas dos matrices diagonalizables \\(\\mathbf{A}\\) y \\(\\mathbf{B}\\), son conmutables (\\(\\mathbf{AB}=\\mathbf{BA}\\)) si y solo si son simult\u00e1neamente diagonalizables (comparten la misma base ortonormal).<\/p>\n<p>Toda matriz \\(\\mathbf{A}\\) de dimensi\u00f3n \\(n\\) y coeficientes reales es diagonalizable si, y s\u00f3lo si, existe una base de \\(\\mathbb{R}^{n}\\) formada por autovectores de \\(\\mathbf{A}\\)<\/p>\n<\/blockquote>\n<p>El resultado anterior nos permite formular la definici\u00f3n de diagonalizaci\u00f3n ortogonal o matriz ortogonalmente diagonalizable: Una matriz cuadrada se dice que es ortogonalmente diagonalizable si y s\u00f3lo si es diagonalizable mediante una matriz de \\(\\mathbf{Q}\\) ortogonal. Por tanto, si una matriz es ortogonalmente diagonalizable si y s\u00f3lo si se puede encontrar una base de \\(\\mathbb{R}^{n}\\) formada por autovectores ortonormales de \\(\\mathbf{A}\\) (que compondr\u00e1n las columnas de la matriz \\(\\mathbf{Q}\\)).<\/p>\n<blockquote>\n<p><strong>Teorema<\/strong>: Una matriz cuadrada y real es ortogonalmente diagonalizable si, y solo si, es sim\u00e9trica.<\/p>\n<\/blockquote>\n<p>Observemos que, aunque podemos hacer ortogonales los autovectores conseguidos, no nos garantiza que la matriz formada por ellos sea ortogonal. Eso solo se producir\u00e1 en el caso de ser una matriz sim\u00e9trica.<\/p>\n<blockquote><p><strong>Ejercicio:<\/strong>Diagonalizar ortogonalmente la matriz<br \/>\n\\[<br \/>\nA=<br \/>\n\\begin{bmatrix}<br \/>\n3 &#038; -2 &#038; 4\\\\<br \/>\n-2 &#038; 6 &#038; 2\\\\<br \/>\n4 &#038; 2 &#038; 3<br \/>\n\\end{bmatrix}.<br \/>\n\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv213g5() {\n  var htmlShow213g5 = document.getElementById(\"html-show213g5\");\n  if (htmlShow213g5.style.display === \"none\") {\n    htmlShow213g5.style.display = \"block\";\n  } else {\n    htmlShow213g5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv213g5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show213g5\" style=\"display: none;\">\nLa matriz es sim\u00e9trica, luego es diagonalizable ortogonalmente.<\/p>\n<p>Los autovalores de la matriz son<br \/>\n\\[<br \/>\n\\lambda_1=7 \\ (\\text{multiplicidad }2),\\qquad \\lambda_2=-2.<br \/>\n\\]<\/p>\n<p>Una base ortonormal de autovectores es:<br \/>\n\\[<br \/>\nu_1=\\frac{1}{\\sqrt2}[1,0,1],\\quad<br \/>\nu_2=\\frac{1}{\\sqrt6}[-1,2,1],\\quad<br \/>\nu_3=\\frac{1}{3}[2,1,-2].<br \/>\n\\]<\/p>\n<p>La matriz ortogonal formada por estos vectores es<br \/>\n\\[<br \/>\nQ=<br \/>\n\\begin{bmatrix}<br \/>\n\\frac{1}{\\sqrt2} &#038; \\frac{1}{\\sqrt6} &#038; -\\frac{2}{3}\\\\<br \/>\n\\frac{1}{\\sqrt2} &#038; -\\frac{1}{\\sqrt6} &#038; \\frac{2}{3}\\\\<br \/>\n0 &#038; \\frac{2}{\\sqrt6} &#038; \\frac{1}{3}<br \/>\n\\end{bmatrix}.<br \/>\n\\]<\/p>\n<p>La matriz diagonal es<br \/>\n\\[<br \/>\nD=\\operatorname{diag}[-2,7,7].<br \/>\n\\]<\/p>\n<p>Por tanto,<br \/>\n\\[<br \/>\nA=QDQ^{T}.<br \/>\n\\]\n<\/p><\/div>\n<hr \/>\n<p>En general, la diagonalizaci\u00f3n ortogonal de una matriz real sim\u00e9trica no es \u00fanica, pero la matriz diagonal resultante s\u00ed lo es salvo por el orden de los autovalores. Lo que no es \u00fanico es la matriz ortogonal de cambio de base, que puede variar dentro de cada subespacio propio.<\/p>\n<h4>Caso con autovalores simples<\/h4>\n<p>Si la matriz sim\u00e9trica tiene todos los autovalores reales y distintos (multiplicidad geom\u00e9trica 1), entonces:<\/p>\n<ul>\n<li>Cada autovalor determina una recta propia, pero hay dos posibles autovectores unitarios en esa recta, \\(\\mathbf{u}\\) y \\(-\\mathbf{u}\\).<\/li>\n<li>Por tanto, la matriz ortogonal \\(\\mathbf{Q}\\) que diagonaliza \\(A\\) es \u00fanica salvo por cambios de signo en sus columnas y permutaci\u00f3n del orden de los autovalores (y columnas asociadas).<\/li>\n<\/ul>\n<h4>Autovalores con multiplicidad mayor<\/h4>\n<p>Si alg\u00fan autovalor tiene multiplicidad geom\u00e9trica mayor que 1, la no unicidad es mayor:<\/p>\n<ul>\n<li>Dentro de cada subespacio propio de dimensi\u00f3n \\(k>1\\) se puede elegir cualquier base ortonormal; todas producen matrices ortogonales \\(Q\\) distintas pero la misma matriz \\(\\mathbf{D}\\) salvo permutaci\u00f3n de bloques.<\/li>\n<li>En consecuencia, la diagonalizaci\u00f3n \\(A = \\mathbf{Q}\\mathbf{D}\\mathbf{Q}^{T}\\) no es \u00fanica: existen infinitas matrices ortogonales \\(\\mathbf{Q}\\) que diagonalizan \\(A\\), todas obtenidas por rotaciones ortogonales independientes dentro de cada subespacio propio.<\/li>\n<\/ul>\n<h4> Qu\u00e9 es realmente \u201c\u00fanico\u201d<\/h4>\n<ul>\n<li>El conjunto de autovalores (con multiplicidad) est\u00e1 completamente determinado por \\(A\\).<\/li>\n<li>La matriz diagonal \\(\\mathbf{D}\\) est\u00e1 determinada salvo reordenar los autovalores en la diagonal.<\/li>\n<li>La matriz ortogonal \\(\\mathbf{Q}\\) nunca es \u00fanica; en el mejor caso, solo es \u00fanica salvo signos y permutaciones de columnas, y en el caso con autovalores m\u00faltiples, tambi\u00e9n salvo cualquier cambio ortogonal dentro de cada subespacio propio.<\/li>\n<\/ul>\n<h3>Potencias de una matriz<\/h3>\n<p>Diagonalizar una matriz nos ayuda a calcular potencias de una matriz \\({\\displaystyle A}\\), si \\({\\displaystyle n\\in \\mathbb {N} }\\) entonces \\[{\\displaystyle A^{n}=PD^{n}P^{-1}}\\]<\/p>\n<blockquote><p><strong>Ejercicio:<\/strong> Determinar \\(\\begin{bmatrix}1&#038;0\\\\6&#038;-1\\end{bmatrix}^{19}\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4r2b() {\n  var htmlShow4r2b = document.getElementById(\"html-show4r2b\");\n  if (htmlShow4r2b.style.display === \"none\") {\n    htmlShow4r2b.style.display = \"block\";\n  } else {\n    htmlShow4r2b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4r2b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4r2b\" style=\"display: none;\">\nPor el ejercicio anterior \\[\\begin{bmatrix}1 &#038; 0\\\\<br \/>\n6 &#038; -1\\end{bmatrix}=\\begin{bmatrix}\\frac{1}{3} &#038; 0\\\\<br \/>\n1 &#038; 1\\end{bmatrix}\\begin{bmatrix}1 &#038; 0\\\\<br \/>\n0 &#038; -1\\end{bmatrix}\\begin{bmatrix}3 &#038; 0\\\\<br \/>\n-3 &#038; 1\\end{bmatrix}.\\]<br \/>\nLuego<br \/>\n\\[\\begin{bmatrix}1 &#038; 0\\\\ 6 &#038; -1\\end{bmatrix}^{19}=\\begin{bmatrix}\\frac{1}{3} &#038; 0\\\\<br \/>\n1 &#038; 1\\end{bmatrix}\\begin{bmatrix}1^{19} &#038; 0\\\\ 0 &#038; (-1)^{19}\\end{bmatrix}\\begin{bmatrix}3 &#038; 0\\\\ -3 &#038; 1\\end{bmatrix}=\\begin{bmatrix}1 &#038; 0\\\\ 6 &#038; -1\\end{bmatrix}\\]\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Determinar \\(\\begin{bmatrix}1&#038;2\\\\2&#038;1\\end{bmatrix}^{9}\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4r2() {\n  var htmlShow4r2 = document.getElementById(\"html-show4r2\");\n  if (htmlShow4r2.style.display === \"none\") {\n    htmlShow4r2.style.display = \"block\";\n  } else {\n    htmlShow4r2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4r2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4r2\" style=\"display: none;\">\nPor ser una matriz sim\u00e9trica y real es diagonalizable por una matriz ortonormal, que ser\u00e1 la dada por los vectores propios unitarios:<br \/>\n\\[\\begin{bmatrix}\\frac{1}{\\sqrt{2}} &#038; -\\frac{1}{\\sqrt{2}}\\\\<br \/>\n\\frac{1}{\\sqrt{2}} &#038; \\frac{1}{\\sqrt{2}}\\end{bmatrix}\\]<br \/>\nLuego \\[\\begin{bmatrix}1 &#038; 2\\\\ 2 &#038; 1\\end{bmatrix}=\\begin{bmatrix}\\frac{1}{\\sqrt{2}} &#038; -\\frac{1}{\\sqrt{2}}\\\\<br \/>\n\\frac{1}{\\sqrt{2}} &#038; \\frac{1}{\\sqrt{2}}\\end{bmatrix} \\begin{bmatrix}3 &#038; 0\\\\ 0 &#038; -1\\end{bmatrix}\\begin{bmatrix}\\frac{1}{\\sqrt{2}} &#038; -\\frac{1}{\\sqrt{2}}\\\\ \\frac{1}{\\sqrt{2}} &#038; \\frac{1}{\\sqrt{2}}\\end{bmatrix}^t\\]<br \/>\nPor tanto,<br \/>\n\\[\\begin{bmatrix}1 &#038; 2\\\\ 2 &#038; 1\\end{bmatrix}^9=\\begin{bmatrix}\\frac{1}{\\sqrt{2}} &#038; -\\frac{1}{\\sqrt{2}}\\\\<br \/>\n\\frac{1}{\\sqrt{2}} &#038; \\frac{1}{\\sqrt{2}}\\end{bmatrix}\\begin{bmatrix}{3}^9 &#038; 0\\\\<br \/>\n0 &#038; (-1)^9\\end{bmatrix}\\begin{bmatrix}\\frac{1}{\\sqrt{2}} &#038; \\frac{1}{\\sqrt{2}}\\\\<br \/>\n-\\frac{1}{\\sqrt{2}} &#038; \\frac{1}{\\sqrt{2}}\\end{bmatrix}=\\begin{bmatrix}9841 &#038; 9842\\\\<br \/>\n9842 &#038; 9841\\end{bmatrix}\\]<\/div>\n<hr \/>\n<h2>Ideas clave para el repaso<\/h2>\n<ul>\n<li>Los valores propios de \\( A \\) son las ra\u00edces reales del polinomio caracter\u00edstico de \\( A \\).<\/li>\n<li>Matrices semejantes(por transformaciones de similaridad) tienen los mismos valores propios.<\/li>\n<li>Una matriz \\( A \\) de \\( n \\times n \\) es diagonalizable si y s\u00f3lo si tiene \\( n \\) vectores propios linealmente independientes. En este caso, \\( A \\) es semejante a una matriz diagonal \\( D \\), con \\( D = P^{-1}AP \\), cuyos elementos en la diagonal son los valores propios de \\( A \\), mientras que \\( P \\) es una matriz cuyas columnas son los \\( n \\) vectores propios linealmente independientes de \\( A \\).<\/li>\n<li>Todas las ra\u00edces del polinomio caracter\u00edstico de una matriz sim\u00e9trica son n\u00fameros reales.<\/li>\n<li>Si \\( A \\) es una matriz sim\u00e9trica, los vectores propios correspondientes a valores propios distintos de \\( A \\) son ortogonales.<\/li>\n<li>La matriz \\( A \\) de \\( n \\times n \\) es ortogonal si y s\u00f3lo si las columnas de \\( A \\) forman un conjunto ortonormal de vectores en \\( \\mathbb{R}^n \\).<\/li>\n<li>Si \\( A \\) es una matriz sim\u00e9trica de \\( n \\times n \\), existe una matriz ortogonal \\( P \\) (\\( P^{-1} = P^t \\)) tal que \\( P^{-1}AP = D \\), una matriz diagonal. Los valores propios de A est\u00e1n sobre la diagonal principal de D.<\/li>\n<\/ul>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\">\n<p><strong>Ejercicio:<\/strong> Sea la matriz \\[A=\\left[\\begin{smallmatrix}0 &amp; -1 &amp; -1 &amp; 0\\\\ -2 &amp; 1 &amp; -1 &amp; 0 \\\\ -2 &amp; 2 &amp; 2 &amp; 0\\\\ 0 &amp; 0 &amp; 0 &amp; -1\\end{smallmatrix}\\right].\\] \u00bfCu\u00e1ntos autovectores tiene?<\/p>\n<div id=\"menu-a\">\n<ul>\n<li>2<\/li>\n<li>3<\/li>\n<li>4<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv() {\n  var htmlShow = document.getElementById(\"html-show\");\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>3.)<\/strong><\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">factor<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }{{\\left( x-2\\right) }^{2}} {{\\left( x+1\\right) }^{2}}=0\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos dos autovalores -1 y 2. Veamos la dimensi\u00f3n de los subespacios propios:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">rA<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rank<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_string\">\u00abdim C_-1=\u00bb<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">rA<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">rA<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rank<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_string\">\u00abdim C_2=\u00bb<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">rA<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\mathbf{dim}\\ C_{-1}=2\\] \\[\\mathbf{dim}\\ C{2}=1\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">En consecuencia, tendremos tres vectores propios.<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Dado \\(\\mathbf {A} \\in M_{n\\times n}(\\mathbb {K} )\\), una matriz cuadrada con valores sobre un cuerpo \\(\\mathbb {K}\\), decimos que \\(\\mathbf{A}\\) es diagonalizable si, y s\u00f3lo si, \\(\\mathbf{A}\\) se puede descomponer de&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-723","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/723","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=723"}],"version-history":[{"count":5,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/723\/revisions"}],"predecessor-version":[{"id":729,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/723\/revisions\/729"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=723"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=723"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=723"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}