{"id":173,"date":"2026-10-14T11:50:56","date_gmt":"2026-10-14T09:50:56","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=173"},"modified":"2026-10-13T18:00:35","modified_gmt":"2026-10-13T16:00:35","slug":"mathbio-autovectores-y-autovalores","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=173","title":{"rendered":"MathBio: Autovectores y autovalores"},"content":{"rendered":"<p>Denominamos esta parte autovectores y autovalores, tambi\u00e9n conocidos como vectores y valores propios de una matriz. Su definici\u00f3n es simple:<\/p>\n<blockquote>\n<p>Dada una matriz, \\(A\\in\\mathcal{M}_n(\\mathbb{R})\\), real porque es el principal cuerpo que trataremos, decimos que \\(\\vec{v} \\in\\mathbb{R}^n\\) es un autovector, o vector propio de la matriz, si \\[A\\vec{v}=\\lambda\\vec{v},\\] siendo \\(\\lambda\\in\\mathbb{R}\\). Al real \\(\\lambda\\) se le denomina autovalor o valor propio de la matriz.<\/p>\n<\/blockquote>\n<p>Nuestro principal prop\u00f3sito es saber determinar los autovectores y autovalores de una matriz real cuadrada.<\/p>\n<p>Para conseguir nuestro prop\u00f3sito necesitamos encontrar las soluciones de la ecuaci\u00f3n que plantea el determinante \\[det(A-\\lambda\\, I),\\] siendo \\(A\\in\\mathcal{M}_n(\\mathbb{R})\\) la matriz cuadrada y \\(I\\) la indentidad en \\(\\mathcal{M}_n(\\mathbb{R})\\).<\/p>\n<p>El polinomio \\(p(\\lambda) = det(A -\\lambda\\,I)\\) es el polinomio caracter\u00edstico de A: los valores propios de una matriz son los ceros de su polinomio caracter\u00edstico (soluciones de la ecuaci\u00f3n caracter\u00edstica).<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovalores de la matriz \\[\\begin{bmatrix}5 &#038; 0 &#038; 4\\\\ 4 &#038; -1 &#038; 4\\\\ -8 &#038; 0 &#038; -7\\end{bmatrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2d() {\n  var htmlShow2d = document.getElementById(\"html-show2d\");\n  if (htmlShow2d.style.display === \"none\") {\n    htmlShow2d.style.display = \"block\";\n  } else {\n    htmlShow2d.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv2d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d\" style=\"display: none;\">\nTenemos que determinar <\/p>\n<p>\\[\\begin{align*}\\mathbf{det}(A-\\lambda I_3)&#038;=\\begin{vmatrix}-\\lambda+5 &#038; 0 &#038; 4\\\\<br \/>\n4 &#038; -\\lambda-1 &#038; 4\\\\<br \/>\n-8 &#038; 0 &#038; -7-\\lambda\\end{vmatrix}\\\\ &#038;=-\\lambda^3-3\\lambda^2+\\lambda+3\\\\ &#038;=(\\lambda+3)(\\lambda+1)(\\lambda-1)\\end{align*}\\]<br \/>\nPor tanto, los autovalores son \\(\\lambda_1=3\\), \\(\\lambda_2=-1\\) y \\(\\lambda_3=1\\).\n<\/p><\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovalores de la matriz \\[\\begin{bmatrix}1 &#038; 1 &#038; 0\\\\<br \/>\n2 &#038; 0 &#038; 2\\\\ -3 &#038; -2 &#038; -1\\end{bmatrix}\\]<\/p>\n<\/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;\">\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_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\">&#8211;<\/span><span class=\"code_number\">3<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/><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\">&#8211;<\/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 \/><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<\/table>\n<p> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable>  <mlabeledtr columnalign=\"left\"> <mtd><mtext>(ecu_caract)<\/mtext> <\/mtd> <mtd><mi>\u2212<\/mi><msup>  <mi>x<\/mi>  <mn>3<\/mn><\/msup><mi>\u2212<\/mi><mi>x<\/mi><mi>=<\/mi><mn>0<\/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><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mi>%i<\/mi><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mi>%i<\/mi><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mn>0<\/mn><mo>]<\/mo> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovalores de la matriz \\[\\begin{bmatrix}4 &#038; 3 &#038; 0\\\\<br \/>\n-6 &#038; -5 &#038; 0\\\\ 3 &#038; 3 &#038; 1\\end{bmatrix}\\]\n<\/p>\n<\/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;\">\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_number\">4<\/span>,<span class=\"code_number\">3<\/span>,<span class=\"code_number\">0<\/span>],[<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">6<\/span>,<span class=\"code_number\">&#8211;<\/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 \/><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\">&#8211;<\/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 \/><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<\/table>\n<p> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable>  <mlabeledtr columnalign=\"left\"> <mtd><mtext>(ecu_caract)<\/mtext> <\/mtd> <mtd><mi>\u2212<\/mi><msup>  <mi>x<\/mi>  <mn>3<\/mn><\/msup><mo>+<\/mo><mn>3<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mi>\u2212<\/mi><mn>2<\/mn><mi>=<\/mi><mn>0<\/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><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mn>2<\/mn><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mn>1<\/mn><mo>]<\/mo> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<p>Cada valor propio tiene asociado un conjunto \\(C_\\lambda=\\{\\vec{v}\\in\\mathbb{R}^n\\}\\), que se determina resolviendo el sistema homogeneo \\((A-\\lambda\\, I)\\vec{x}=\\vec{0}\\). Las soluciones de estos sistemas ser\u00e1n los vectores propios de la matriz, que veremos m\u00e1s adelante.<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovectores del autovalor \\(\\lambda=-3\\) de la matriz \\[\\begin{bmatrix}5 &#038; 0 &#038; 4\\\\ 4 &#038; -1 &#038; 4\\\\ -8 &#038; 0 &#038; -7\\end{bmatrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2d45() {\n  var htmlShow2d45 = document.getElementById(\"html-show2d45\");\n  if (htmlShow2d45.style.display === \"none\") {\n    htmlShow2d45.style.display = \"block\";\n  } else {\n    htmlShow2d45.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv2d45()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d45\" style=\"display: none;\">\nPara calcular los autovectores del autovalor \\(\\lambda=-3\\), planteamos el sistema<\/p>\n<p>\\[(A-\\lambda I_3)X=0\\to \\begin{bmatrix}-3+5 &#038; 0 &#038; 4\\\\<br \/>\n4 &#038; -3-1 &#038; 4\\\\<br \/>\n-8 &#038; 0 &#038; -7-3\\end{bmatrix}\\begin{bmatrix}x\\\\ y\\\\ z\\end{bmatrix}=\\begin{bmatrix}0\\\\ 0\\\\ 0\\end{bmatrix}\\]<\/p>\n<p>Utilicemos las matrices como hecho en casos anteriores para resolver este sistema:<br \/>\n\\[\\begin{bmatrix}8 &#038; 0 &#038; 4\\\\<br \/>\n4 &#038; 2 &#038; 4\\\\<br \/>\n-8 &#038; 0 &#038; -4\\end{bmatrix}\\sim<br \/>\n\\begin{bmatrix}8 &#038; 0 &#038; 4\\\\<br \/>\n4 &#038; 2 &#038; 4\\\\<br \/>\n0 &#038; 0 &#038; 0\\end{bmatrix}\\sim<br \/>\n\\begin{bmatrix}2 &#038; 0 &#038; 1\\\\<br \/>\n2 &#038; 1 &#038; 2\\\\<br \/>\n0 &#038; 0 &#038; 0\\end{bmatrix}\\sim<br \/>\n\\begin{bmatrix}2 &#038; 0 &#038; 1\\\\<br \/>\n0 &#038; 1 &#038; 1\\\\<br \/>\n0 &#038; 0 &#038; 0\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>Luego, \\(x=-\\frac{1}{2}\\mu\\), \\(y=-\\mu\\) y \\(z=\\mu\\), por tanto el subespacio asociado al autovalor \\(\\lambda=-3\\) es \\(\\mathcal{C}_{-3}=\\mathbf{Gen}\\{[-1,-2,2]\\}\\) <\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovectores del autovalor \\(\\lambda=-1\\) de la matriz \\[\\begin{bmatrix}5 &#038; 0 &#038; 4\\\\ 4 &#038; -1 &#038; 4\\\\ -8 &#038; 0 &#038; -7\\end{bmatrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2d4() {\n  var htmlShow2d4 = document.getElementById(\"html-show2d4\");\n  if (htmlShow2d4.style.display === \"none\") {\n    htmlShow2d4.style.display = \"block\";\n  } else {\n    htmlShow2d4.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv2d4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d4\" style=\"display: none;\">\nPara calcular los autovectores del autovalor \\(\\lambda=-1\\), planteamos el sistema<\/p>\n<p>\\[(A-\\lambda I_3)X=0\\to \\begin{bmatrix}1+5 &#038; 0 &#038; 4\\\\<br \/>\n4 &#038; -1+1 &#038; 4\\\\<br \/>\n-8 &#038; 0 &#038; -7+1\\end{bmatrix}\\begin{bmatrix}x\\\\ y\\\\ z\\end{bmatrix}=\\begin{bmatrix}0\\\\ 0\\\\ 0\\end{bmatrix}\\]<\/p>\n<p>Utilicemos las matrices como hecho en casos anteriores para resolver este sistema:<br \/>\n\\[\\begin{bmatrix}6 &#038; 0 &#038; 4\\\\<br \/>\n4 &#038; 0 &#038; 4\\\\<br \/>\n-8 &#038; 0 &#038; -6\\end{bmatrix}\\sim<br \/>\n\\begin{bmatrix}3 &#038; 0 &#038; 2\\\\<br \/>\n1 &#038; 0 &#038; 1\\\\<br \/>\n-4 &#038; 0 &#038; -3\\end{bmatrix}\\sim<br \/>\n\\begin{bmatrix}0 &#038; 0 &#038; -1\\\\<br \/>\n1 &#038; 0 &#038; 1\\\\<br \/>\n-4 &#038; 0 &#038; -3\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>Luego, \\(x=0\\) y \\(z=0\\), por tanto el subespacio asociado al autovalor \\(\\lambda=-1\\) es \\(\\mathcal{C}_{-1}=\\mathbf{Gen}\\{[0,1,0]\\}\\) <\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovectores del autovalor \\(\\lambda=1\\) de la matriz \\[\\begin{bmatrix}5 &#038; 0 &#038; 4\\\\ 4 &#038; -1 &#038; 4\\\\ -8 &#038; 0 &#038; -7\\end{bmatrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2d4w() {\n  var htmlShow2d4w = document.getElementById(\"html-show2d4w\");\n  if (htmlShow2d4w.style.display === \"none\") {\n    htmlShow2d4w.style.display = \"block\";\n  } else {\n    htmlShow2d4w.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv2d4w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d4w\" style=\"display: none;\">\nPara calcular los autovectores del autovalor \\(\\lambda=1\\), planteamos el sistema<\/p>\n<p>\\[(A-\\lambda I_3)X=0\\to \\begin{bmatrix}5-1 &#038; 0 &#038; 4\\\\<br \/>\n4 &#038; -1-1 &#038; 4\\\\<br \/>\n-8 &#038; 0 &#038; -7-1\\end{bmatrix}\\begin{bmatrix}x\\\\ y\\\\ z\\end{bmatrix}=\\begin{bmatrix}0\\\\ 0\\\\ 0\\end{bmatrix}\\]<\/p>\n<p>Utilicemos las matrices como hecho en casos anteriores para resolver este sistema:<br \/>\n\\[\\begin{bmatrix}4 &#038; 0 &#038; 4\\\\<br \/>\n4 &#038; -2 &#038; 4\\\\<br \/>\n-8 &#038; 0 &#038; -8\\end{bmatrix}\\sim<br \/>\n\\begin{bmatrix}4 &#038; 0 &#038; 4\\\\<br \/>\n0 &#038; -2 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0\\end{bmatrix}\\sim<br \/>\n\\begin{bmatrix}1 &#038; 0 &#038; 1\\\\<br \/>\n0 &#038; 1 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0\\end{bmatrix}<br \/>\n\\]<\/p>\n<p>Luego, \\(y=0\\) y \\(x=-z\\), por tanto el subespacio asociado al autovalor \\(\\lambda=1\\) es \\(\\mathcal{C}_{1}=\\mathbf{Gen}\\{[-1,0,1]\\}\\) <\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determinar los autovectores de la matriz \\[\\begin{bmatrix}4 &#038; 0 &#038; -3\\\\<br \/>\n3 &#038; -1 &#038; 3\\\\<br \/>\n6 &#038; 0 &#038; 5\\end{bmatrix}\\] correspondientes a su autovalor real.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2b() {\n  var htmlShow2b = document.getElementById(\"html-show2b\");\n  if (htmlShow2b.style.display === \"none\") {\n    htmlShow2b.style.display = \"block\";\n  } else {\n    htmlShow2b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2b\" 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_number\">4<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>],[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">3<\/span>],[<span class=\"code_number\">6<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">5<\/span>])<span class=\"code_endofline\">$<\/span><br \/><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\">&#8211;<\/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 \/><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<\/table>\n<p> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(ecu_caract)<\/mtext> <\/mtd> <mtd><mi>\u2212<\/mi><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>+<\/mo><mn>8<\/mn><mo>\u2062<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>\u2212<\/mi><mn>29<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mi>\u2212<\/mi><mn>38<\/mn><mi>=<\/mi><mn>0<\/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><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mfrac><mrow> <msqrt><mn>71<\/mn> <\/msqrt> <mo>\u2062<\/mo> <mi>%i<\/mi> <mi>\u2212<\/mi> <mn>9<\/mn><\/mrow><mn>2<\/mn><\/mfrac><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mfrac><mrow> <msqrt><mn>71<\/mn> <\/msqrt> <mo>\u2062<\/mo> <mi>%i<\/mi> <mo>+<\/mo> <mn>9<\/mn><\/mrow><mn>2<\/mn><\/mfrac><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mn>1<\/mn><mo>]<\/mo> <\/mtd><\/mlabeledtr><\/mtable> <\/math><\/p>\n<p>El \u00fanico autovalor real es \\(\\lambda=-1\\), luego el autovector ser\u00e1 la base del subespacio dado por \\[\\begin{bmatrix}5 &#038; 0 &#038; -3\\\\ 3 &#038; 0 &#038; 3\\\\ 6 &#038; 0 &#038; 6\\end{bmatrix}\\begin{bmatrix}x \\\\ y\\\\ z\\end{bmatrix}=\\begin{bmatrix}0 \\\\ 0\\\\ 0\\end{bmatrix}\\]<\/p>\n<p>Mediante operaciones elementales, el sistema anterior es equivalente a \\[\\begin{bmatrix}1 &#038; 0 &#038; 0\\\\<br \/>\n1 &#038; 0 &#038; 1\\\\ 0 &#038; 0 &#038; 0\\end{bmatrix}\\begin{bmatrix}x \\\\ y\\\\ z\\end{bmatrix}=\\begin{bmatrix}0 \\\\ 0\\\\ 0\\end{bmatrix}\\]<\/p>\n<p>Luego, \\(x=0\\) y \\(z=0\\), por tanto el subespacio asociado al autovalor \\(\\lambda=-1\\) es \\(\\mathcal{C}_{-1}=\\mathbf{Gen}\\{[0,1,0]\\}\\) <\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dados los autovectores de la matriz \\[\\begin{bmatrix}-4 &#038; 0 &#038; -3\\\\<br \/>\n3 &#038; -1 &#038; 3\\\\<br \/>\n6 &#038; 0 &#038; 5\\end{bmatrix}\\] del mayor autovalor real, \u00bfcu\u00e1l es el producto escalar de [1,1,1] por uno de dichos vectores unitarios?.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2c() {\n  var htmlShow2c = document.getElementById(\"html-show2c\");\n  if (htmlShow2c.style.display === \"none\") {\n    htmlShow2c.style.display = \"block\";\n  } else {\n    htmlShow2c.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2c()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2c\" 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_number\">-4<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>],[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">3<\/span>],[<span class=\"code_number\">6<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">5<\/span>])<span class=\"code_endofline\">$<\/span><br \/><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\">&#8211;<\/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 \/><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<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(ecu_caract)<\/mtext> <\/mtd> <mtd><mi>\u2212<\/mi><msup><mi>x<\/mi><mn>3<\/mn><\/msup> <mo>+<\/mo><mn>3<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>=<\/mi><mn>0<\/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><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mn>2<\/mn><mo><mo>,<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mn>1<\/mn><mo>]<\/mo> <\/mtd><\/mlabeledtr><\/mtable> <\/math><\/p>\n<p>El autovalor real mayor es \\(\\lambda=2\\), luego los autovectores ser\u00e1n la base del subespacio dado por \\[\\begin{bmatrix}-6 &#038; 0 &#038; -3\\\\ 3 &#038; -3 &#038; 3\\\\ 6 &#038; 0 &#038; 3\\end{bmatrix}\\begin{bmatrix}x \\\\ y\\\\ z\\end{bmatrix}=\\begin{bmatrix}0 \\\\ 0\\\\ 0\\end{bmatrix}\\]<\/p>\n<p>Mediante operaciones elementales, el sistema anterior es equivalente a \\[\\begin{bmatrix}1 &#038; 0 &#038; \\frac{1}{2}\\\\<br \/>\n0 &#038; -1 &#038; \\frac{1}{2}\\\\ 0 &#038; 0 &#038; 0\\end{bmatrix}\\begin{bmatrix}x \\\\ y\\\\ z\\end{bmatrix}=\\begin{bmatrix}0 \\\\ 0\\\\ 0\\end{bmatrix}\\]<\/p>\n<p>Luego, \\(x=\\mu\\frac{-1}{2}\\), \\(y=\\mu\\frac{1}{2}\\) y \\(z=\\mu\\), por tanto, el subespacio asociado al autovalor \\(\\lambda=2\\) es \\(\\mathcal{C}_{2}=\\mathbf{Gen}\\{[-1,1,2]\\}\\)<\/p>\n<p>Ahora solo tenemos que calcular \\[\\frac{1}{\\|[-1,1,2]\\|}([1,1,1]\\bullet[-1,1,2])=\\frac{\\sqrt{6}}{3}\\]<\/p>\n<\/div>\n<hr \/>\n<p>Pod\u00e9is ver m\u00e1s ejemplos en <a href=\"http:\/\/en.wikibooks.org\/wiki\/Linear_Algebra\/Eigenvalues_and_Eigenvectors\" target=\"_blank\" rel=\"noopener noreferrer\">Linear Algebra\/Eigenvalues and Eigenvectors<\/a>.<\/p>\n<hr \/>\n<section class=\"al-aplicaciones\" aria-labelledby=\"al-titulo\">\n<div class=\"al-cierre\">\n<h2>Aplicaci\u00f3n<\/h2>\n<p>A continuaci\u00f3n, un documento con el planteamiento de un problema aplicado a la biotecnolog\u00eda (Din\u00e1mica Poblacional en Fotobiorreactores), donde buscamos predecir la densidad poblacional total tras \\(k = 5\\) d\u00edas de cultivo de una poblaci\u00f3n de microalgas, junto con su soluci\u00f3n detallada paso a paso:<\/p>\n<p>&nbsp;<\/p>\n<blockquote>\n<ul>\n<li><a href=\"https:\/\/uploads.jesussoto.es\/mathbio\/Mathbio_problema_autovalores.html\" style=\"color: #0066cc;\" target=\"_blank\">Autovalores, Autovectores y Din\u00e1mica Poblacional en Fotobiorreactores<\/a><\/li>\n<\/ul>\n<\/blockquote><\/div>\n<\/section>\n<hr \/>\n<h3>Bibliograf\u00eda<\/h3>\n<ul>\n<li>Cap\u00edtulo 5 de \u00c1lgebra lineal y sus aplicaciones. 5\u00ba edici\u00f3n, David C. Lay. Pearson. 2016.<\/li>\n<\/ul>\n<hr \/>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\"><strong>Ejercicio:<\/strong>  \u00bfCu\u00e1ntos autovalores distintos tiene la matriz \\(A=\\begin{bmatrix}2&#038;1&#038;0\\\\ 1&#038;2&#038;0 \\\\ 0&#038;0&#038;1\\end{bmatrix}\\)?<\/p>\n<div id=\"menu-a\">\n<ul>\n<li>1<\/li>\n<li>2<\/li>\n<li>3<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv(id) {\n  var htmlShow = document.getElementById(id);\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<p><em>La soluci\u00f3n se mostrar\u00e1 m\u00e1s adelante.<\/em><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Denominamos esta parte autovectores y autovalores, tambi\u00e9n conocidos como vectores y valores propios de una matriz. Su definici\u00f3n es simple: Dada una matriz, \\(A\\in\\mathcal{M}_n(\\mathbb{R})\\), real porque es el principal cuerpo que trataremos,&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-173","post","type-post","status-publish","format-standard","hentry","category-mathbio"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/173","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=173"}],"version-history":[{"count":6,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/173\/revisions"}],"predecessor-version":[{"id":324,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/173\/revisions\/324"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}