{"id":714,"date":"2025-12-12T10:20:21","date_gmt":"2025-12-12T09:20:21","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=714"},"modified":"2025-12-09T11:00:06","modified_gmt":"2025-12-09T10:00:06","slug":"alg-autovectores-y-autovalores-con-maxima-2","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=714","title":{"rendered":"ALG: Autovectores y autovalores con maxima"},"content":{"rendered":"<p>El pasado d\u00eda vimos que para calcular los valores propios o autovalores necesitamos el polinomio caracter\u00edstico.<\/p>\n<blockquote><p><strong>Ejercicio:<\/strong> Determinar los autovalores de la matriz \\[\\begin{bmatrix}1 &amp; 1 &amp; 0\\\\<br \/>\n2 &amp; 0 &amp; 2\\\\ -3 &amp; -2 &amp; -1\\end{bmatrix}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2() {\n  var htmlShow2 = document.getElementById(\"html-show2\");\n  if (htmlShow2.style.display === \"none\") {\n    htmlShow2.style.display = \"block\";\n  } else {\n    htmlShow2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2\" style=\"display: none;\"><!-- Code cell --><\/p>\n<table>\n<tbody>\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_number\">1<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">2<\/span>],[<span class=\"code_number\">\u2013<\/span><span class=\"code_number\">3<\/span>,<span class=\"code_number\">\u2013<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">\u2013<\/span><span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/>\n<span class=\"code_variable\">ecu_caract<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">expand<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2013<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_function\">ident<\/span>(<span class=\"code_number\">3<\/span>)))<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><br \/>\n<span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">ecu_caract<\/span>,[<span class=\"code_variable\">x<\/span>])<span class=\"code_endofline\">;<\/span> <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>(ecu_caract)\u2212x3\u2212x=0(%o3) [x=\u2212%i,x=%i,x=0]\n<\/p><\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Determinar los autovalores de la matriz \\[\\begin{bmatrix}4 &amp; 3 &amp; 0\\\\<br \/>\n-6 &amp; -5 &amp; 0\\\\ 3 &amp; 3 &amp; 1\\end{bmatrix}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2a() {\n  var htmlShow2a = document.getElementById(\"html-show2a\");\n  if (htmlShow2a.style.display === \"none\") {\n    htmlShow2a.style.display = \"block\";\n  } else {\n    htmlShow2a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a\" style=\"display: none;\"><!-- Code cell --><\/p>\n<table>\n<tbody>\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_number\">4<\/span>,<span class=\"code_number\">3<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">\u2013<\/span><span class=\"code_number\">6<\/span>,<span class=\"code_number\">\u2013<\/span><span class=\"code_number\">5<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">3<\/span>,<span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/>\n<span class=\"code_variable\">ecu_caract<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">expand<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2013<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_function\">ident<\/span>(<span class=\"code_number\">3<\/span>)))<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><br \/>\n<span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">ecu_caract<\/span>,[<span class=\"code_variable\">x<\/span>])<span class=\"code_endofline\">;<\/span> <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>(ecu_caract)\u2212x3+3\u2062x\u22122=0(%o3) [x=\u22122,x=1]\n<\/p><\/div>\n<hr \/>\n<p>Recodad que defin\u00edamos los autovectores, o vectores propios, como<\/p>\n<blockquote>\n<p>Recordemos que dada una matriz, \\(\\mathbf{A}\\in\\mathcal{M}_n(\\mathbb{K})\\), decimos que \\(\\vec{v} \\in\\mathbb{K}^n\\) es un autovector, o vector propio de la matriz, s\u00ed \\[\\mathbf{A}\\vec{v}=\\lambda\\vec{v},\\] siendo \\(\\lambda\\in\\mathbb{K}\\). Al escalar \\(\\lambda\\) se le denomina autovalor o valor propio de la matriz.<\/p>\n<\/blockquote>\n<p>Si determinamos los valores propios, o autovalores, saber los vectores propios, o autovalores, ser\u00e1 equivalente a resolver el sistema \\[(\\mathbf{A}-\\lambda\\, I)\\vec{x}=\\vec{0},\\] para cada valor propio \\(\\lambda\\) ra\u00edz de la ecuaci\u00f3n caracter\u00edstica(el resultado de igualar el polinomio caracter\u00edstico a cero). As\u00ed, para cada valor propio habr\u00e1 un vector propio, al menos.<\/p>\n<p>Cada valor propio tiene asociado un conjunto \\(C_\\lambda=\\{\\vec{v}\\in\\mathbb{K}^n\\}\\), que se determina resolviendo el sistema homogeneo \\((\\mathbf{A}-\\lambda\\, I)\\vec{x}=\\vec{0}\\), que se denomina subespacio propio de \\(\\lambda\\). Las bases de estos subespacios son los vectores propios de la matriz.<\/p>\n<blockquote><p><strong>Ejercicio:<\/strong> Determinar los subespacios propios de la matriz \\[\\begin{bmatrix}2&#038;1\\\\1&#038;2\\end{bmatrix}\\]<\/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<!-- 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\">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_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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_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_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\"> do <\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">aut<\/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\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s<\/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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{{\\left( 2-x\\right) }^{2}}-1\\]<\/p>\n<p>\\[\\left[ x=3,x=1\\right] \\]<\/p>\n<p>\\[\\left[ 3,1\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos dos autovalores, veamos cuales son los subespacios propios de cada autovalor.<\/p>\n<p>Para el primer autovalor<\/p><\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/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_variable\">aut<\/span><span class=\"code_operator\">[<\/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\">v<\/span><span class=\"code_operator\">:<\/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_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}y-x\\\\x-y\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (2)\\]<\/p>\n<p>\\[\\left[ x={\\mathrm{\\% r1}},y={\\mathrm{\\% r1}}\\right] \\]<\/p>\n<p>\\[\\left[ \\left[ 1,1\\right] \\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Para el segundo autovalor.<\/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_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">aut<\/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\">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\">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_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_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_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}y+x\\\\y+x\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (2)\\]<\/p>\n<p>\\[\\left[ x=-{\\mathrm{\\% r2}},y={\\mathrm{\\% r2}}\\right] \\]<\/p>\n<p>\\[\\left[ \\left[ 1,1\\right] ,\\left[ -1,1\\right] \\right] \\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Determinar los subespacios propios de la matriz \\[\\begin{bmatrix}3&amp;2&amp;{-1}\\\\2&amp;3&amp;1\\\\0&amp;0&amp;5\\end{bmatrix}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2g() {\n  var htmlShow2g = document.getElementById(\"html-show2g\");\n  if (htmlShow2g.style.display === \"none\") {\n    htmlShow2g.style.display = \"block\";\n  } else {\n    htmlShow2g.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2g()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2g\" 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\">(%i6)<\/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\">3<\/span><span class=\"code_endofline\">,<\/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_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\">3<\/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\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_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_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\"> do <\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">aut<\/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\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s<\/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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-{{x}^{3}}+11 {{x}^{2}}-35 x+25=0\\]<\/p>\n<p>\\[\\left[ x=1,x=5\\right] \\]<\/p>\n<p>\\[\\left[ 1,5\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos dos autovalores, veamos cuales son los subespacios propios de cada autovalor.<\/p>\n<p>Para el primer autovalor<\/p><\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">aut<\/span><span class=\"code_operator\">[<\/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\">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><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_endofline\">,<\/span><span class=\"code_variable\">%<\/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><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/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_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}-z+2 y+2 x\\\\z+2 y+2 x\\\\4 z\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (1)\\]<\/p>\n<p>\\[\\left[ x={\\mathrm{\\% r1}},y=-{\\mathrm{\\% r1}},z=0\\right] \\]<\/p>\n<p>\\[\\left[ \\left[ 1,-1,0\\right] \\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Para el segundo autovalor.<\/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\">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_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">aut<\/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\">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><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_endofline\">,<\/span><span class=\"code_variable\">%<\/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><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}-z+2 y-2 x\\\\z-2 y+2 x\\\\0\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (3 2)\\]<\/p>\n<p>\\[\\left[ x=-\\frac{{\\mathrm{\\% r2}}-2 {\\mathrm{\\% r3}}}{2},y={\\mathrm{\\% r3}},z={\\mathrm{\\% r2}}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos dos par\u00e1metros: saldr\u00e1n dos vectores 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\">(%i15) <\/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_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_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_endofline\">,<\/span><span class=\"code_variable\">%rnum_list<\/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_number\">0<\/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_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\">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_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_variable\">z<\/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\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%rnum_list<\/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_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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\left[ 1,-1,0\\right] ,\\left[ -\\frac{1}{2},0,1\\right] ,\\left[ 1,1\\right] \\right] \\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Cu\u00e1ntos vectores propio tiene la matriz \\[\\begin{bmatrix}4 &amp; -1 &amp; 6 &amp; 0\\\\<br \/>\n2 &amp; 1 &amp; 6 &amp; 0\\\\<br \/>\n2 &amp; -1 &amp; 8 &amp; 0\\\\<br \/>\n0 &amp; 2 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2g2() {\n  var htmlShow2g2 = document.getElementById(\"html-show2g2\");\n  if (htmlShow2g2.style.display === \"none\") {\n    htmlShow2g2.style.display = \"block\";\n  } else {\n    htmlShow2g2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2g2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2g2\" 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\">(%i6)<\/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\">4<\/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\">6<\/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_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/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\">8<\/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\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/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\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_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_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\"> do <\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span>  <span class=\"code_variable\"> &nbsp; &nbsp;   aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">aut<\/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\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s<\/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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{{x}^{4}}-14 {{x}^{3}}+53 {{x}^{2}}-76 x+36=0\\]<\/p>\n<p>\\[\\left[ x=9,x=1,x=2\\right] \\]<\/p>\n<p>\\[\\left[ 9,1,2\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos tres autovalores, veamos cuales son los subespacios propios de cada autovalor.<\/p>\n<p>Para el primer autovalor<\/p><\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">t<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/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_variable\">aut<\/span><span class=\"code_operator\">[<\/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_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_endofline\">,<\/span><span class=\"code_variable\">%<\/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_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/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_endofline\">,<\/span><span class=\"code_variable\">t<\/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_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_endofline\">,<\/span><span class=\"code_variable\">t<\/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_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}6 z-y-5 x\\\\6 z-8 y+2 x\\\\-z-y+2 x\\\\2 y-8 t\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (1)\\]<\/p>\n<p>\\[\\left[ x={\\mathrm{\\% r1}},y={\\mathrm{\\% r1}},z={\\mathrm{\\% r1}},t=\\frac{{\\mathrm{\\% r1}}}{4}\\right] \\]<\/p>\n<p>\\[\\left[ \\left[ 1,1,1,\\frac{1}{4}\\right] \\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Para el segundo autovalor.<\/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\">(%i12) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">aut<\/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_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_endofline\">,<\/span><span class=\"code_variable\">%<\/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_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/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_endofline\">,<\/span><span class=\"code_variable\">t<\/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}6 z-y+3 x\\\\6 z+2 x\\\\7 z-y+2 x\\\\2 y\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (1)\\]<\/p>\n<p>\\[\\left[ x=0,y=0,z=0,t={\\mathrm{\\% r2}}\\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\">(%i13) <\/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_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_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_endofline\">,<\/span><span class=\"code_variable\">t<\/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_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_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\left[ 1,1,1,\\frac{1}{4}\\right] ,\\left[ 0,0,0,1\\right] \\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Para el tercer autovalor.<\/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\">(%i15) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">aut<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/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_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_endofline\">,<\/span><span class=\"code_variable\">%<\/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_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/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_endofline\">,<\/span><span class=\"code_variable\">t<\/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}6 z-y+2 x\\\\6 z-y+2 x\\\\6 z-y+2 x\\\\2 y-t\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (3 1)\\]<\/p>\n<p>\\[\\left[ x={\\mathrm{\\% r3}},y=6 {\\mathrm{\\% r4}}+2 {\\mathrm{\\% r3}},z={\\mathrm{\\% r4}},t=12 {\\mathrm{\\% r4}}+4 {\\mathrm{\\% r3}}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos dos par\u00e1metros: saldr\u00e1n dos vectores 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\">(%i17) <\/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_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_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_endofline\">,<\/span><span class=\"code_variable\">t<\/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_endofline\">,<\/span><span class=\"code_variable\">%rnum_list<\/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_number\">0<\/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_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\">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_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_endofline\">,<\/span><span class=\"code_variable\">t<\/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\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">%rnum_list<\/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_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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\left[ 1,1,1,\\frac{1}{4}\\right] ,\\left[ 0,0,0,1\\right] ,\\left[ 1,2,0,4\\right] ,\\left[ 0,6,1,12\\right] \\right] \\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Cu\u00e1l es el producto escalar de [1,-2,3,-4] por el autovector unitario del mayor valor propio de la matriz \\[\\begin{bmatrix}4 &amp; 2 &amp; 0 &amp; 0 \\\\<br \/>\n3 &amp; 3 &amp; 0 &amp; 0 \\\\<br \/>\n0 &amp; 0 &amp; 2 &amp; 5 \\\\<br \/>\n0 &amp; 0 &amp; 0 &amp; 2\\end{bmatrix}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2g3() {\n  var htmlShow2g3 = document.getElementById(\"html-show2g3\");\n  if (htmlShow2g3.style.display === \"none\") {\n    htmlShow2g3.style.display = \"block\";\n  } else {\n    htmlShow2g3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2g3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2g3\" 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\">(%i6)<\/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\">4<\/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_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\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/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\">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_endofline\">,<\/span><span class=\"code_number\">5<\/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_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\">rat<\/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_number\">0<\/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\">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 class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">for<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span><span class=\"code_function\">thru<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\">do<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">aut<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{{x}^{4}}-11 {{x}^{3}}+38 {{x}^{2}}-52 x+24=0\\]<\/p>\n<p>\\[\\left[ x=6,x=1,x=2\\right] \\]<\/p>\n<p>\\[\\left[ 6,1,2\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos tres autovalores y el mayor es 6. 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\">(%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_endofline\">,<\/span><span class=\"code_operator\">[<\/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_variable\">t<\/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\">6<\/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_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_endofline\">,<\/span><span class=\"code_variable\">%<\/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_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/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_endofline\">,<\/span><span class=\"code_variable\">t<\/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_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/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_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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}2 y-2 x\\\\3 x-3 y\\\\5 t-4 z\\\\-4 t\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (1)\\]<\/p>\n<p>\\[\\left[ x={\\mathrm{\\% r1}},y={\\mathrm{\\% r1}},z=0,t=0\\right] \\]<\/p>\n<p>\\[\\left[ 1,1,0,0\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El producto que nos piden 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\">(%i12) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><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\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/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_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-\\frac{1}{\\sqrt{2}}\\]<\/p>\n<p>\\[-0.7071067811865475\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejercicio:<\/strong> Cu\u00e1l es el producto escalar de [1,-2,3,-4] por el autovector unitario del mayor valor propio de la matriz \\[\\begin{bmatrix}4 &#038; 1 &#038; 0 &#038; 1\\\\<br \/>\n2 &#038; 3 &#038; 0 &#038; 1\\\\<br \/>\n-2 &#038; 1 &#038; 2 &#038; -3\\\\<br \/>\n2 &#038; -1 &#038; 0 &#038; 5\\end{bmatrix}\\]<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2g3a() {\n  var htmlShow2g3a = document.getElementById(\"html-show2g3a\");\n  if (htmlShow2g3a.style.display === \"none\") {\n    htmlShow2g3a.style.display = \"block\";\n  } else {\n    htmlShow2g3a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2g3a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2g3a\" 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\">(%i6)<\/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\">4<\/span><span class=\"code_endofline\">,<\/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\">1<\/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\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/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_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_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/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_endofline\">,<\/span><span class=\"code_number\">5<\/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\">rat<\/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_number\">0<\/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\">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 class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">for<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">1<\/span><span class=\"code_function\">thru<\/span><span class=\"code_function\">length<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">)<\/span><span class=\"code_function\">do<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_variable\">aut<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">aut<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">s<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">aut<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{{x}^{4}}-14 {{x}^{3}}+68 {{x}^{2}}-136 x+96=0\\]<\/p>\n<p>\\[\\left[ x=6,x=4,x=2\\right] \\]<\/p>\n<p>\\[\\left[ 6,4,2\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Tenemos tres autovalores y el mayor es 6. 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\">(%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_endofline\">,<\/span><span class=\"code_operator\">[<\/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_variable\">t<\/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\">6<\/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_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_endofline\">,<\/span><span class=\"code_variable\">%<\/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_endofline\">,<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/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_endofline\">,<\/span><span class=\"code_variable\">t<\/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_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">t<\/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_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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}y-2 x+t\\\\-3 y+2 x+t\\\\-4 z+y-2 x-3 t\\\\-y+2 x-t\\end{bmatrix}\\]\\[solve: dependent equations eliminated: (4)\\]<\/p>\n<p>\\[\\left[ x={\\mathrm{\\% r1}},y={\\mathrm{\\% r1}},z=-{\\mathrm{\\% r1}},t={\\mathrm{\\% r1}}\\right] \\]<\/p>\n<p>\\[\\left[ 1,1,-1,1\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El producto que nos piden 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_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/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_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-4\\]<\/p>\n<\/div>\n<hr \/>\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>Cu\u00e1ntos suman los coeficientes del polinomio caracter\u00edstico de la matriz \\[\\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].\\] <\/p>\n<div id=\"menu-a\">\n<ul>\n<li>1<\/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>C.)<\/strong><\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Calculemos el polinomio caracter\u00edstico.<\/p>\n<\/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_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">0<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">0<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">define<\/span>(<span class=\"code_function\">f<\/span>(<span class=\"code_variable\">x<\/span>),<span class=\"code_function\">expand<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_variable\">A<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">*<\/span><span class=\"code_function\">ident<\/span>(<span class=\"code_number\">4<\/span>))))<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">f<\/span>(<span class=\"code_number\">1<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o2) <\/mtext><\/mtd><mtd><mrow><mi>f<\/mi><mo>\u2061<\/mo><mrow><mo>(<\/mo><mi>x<\/mi><mo>)<\/mo><\/mrow><\/mrow><mo>:=<\/mo><msup><mi>x<\/mi><mn>4<\/mn><\/msup><mi>\u2212<\/mi><mn>2<\/mn><mo>\u2062<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mi>\u2212<\/mi><mn>3<\/mn><mo>\u2062<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o3) <\/mtext><\/mtd><mtd><mn>4<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>El pasado d\u00eda vimos que para calcular los valores propios o autovalores necesitamos el polinomio caracter\u00edstico. Ejercicio: Determinar los autovalores de la matriz \\[\\begin{bmatrix}1 &amp; 1 &amp; 0\\\\ 2 &amp; 0 &amp;&#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":[7],"class_list":["post-714","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-practicas-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/714","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=714"}],"version-history":[{"count":4,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/714\/revisions"}],"predecessor-version":[{"id":718,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/714\/revisions\/718"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=714"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=714"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=714"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}