ALG12-Eigenvalues_Eigenvectors_COMPLETO.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Autovalores y Autovectores | Eigenvalues, Eigenvectores\n",
    "\n",
    "Ya vimos anteriormente que un sistema de ecuaciones lineales tiene la siguiente forma:<br><br>\n",
    "\n",
    "<center>$Ax=B$</center>\n",
    "\n",
    "¿De dónde proviene los **autovalores** y **autovectores**? Pues provienen de realizarse la siguiente pregunta:\n",
    "\n",
    "<div class=\"alert alert-success\">\n",
    "Suponiendo que tenemos una matriz cuadrada $A$ de $nxn$, ¿existe algún vector $x$ distinto de $0$ para el cual $Ax$ es un escalar múltiplo de $x$? Si escribimos esta pregunta en lenguaje matemático, nos encontramos con la definición de los autovalores y autovectores:<br><br>\n",
    "    \n",
    "<center>$Ax = \\lambda x$</center>\n",
    "</div>\n",
    "\n",
    "Cuando esta ecuación es válida y $x$ no es $0$, decimos que $\\lambda$ es el **Eigenvalue** o **autovalores** de $A$ Y $x$ es su correspondiente **Eigenvector** o **autovector**. Un ejemplo muy simple es el siguiente:<br><br>\n",
    "\n",
    "$Ax = \\lambda x$\n",
    "\n",
    "<center>$A\\left( {\\begin{array}{cc}\n",
    "   1\\\\\n",
    "   1\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   5\\\\\n",
    "   5\\\\\n",
    "  \\end{array} } \\right) = 5\\left( {\\begin{array}{cc}\n",
    "   1\\\\\n",
    "   1\\\\\n",
    "  \\end{array} } \\right)$ </center><br><br>\n",
    "  \n",
    "En este caso:\n",
    "\n",
    "$\\lambda = 5$<br>\n",
    "$x = \\left( {\\begin{array}{cc}\n",
    "   1\\\\\n",
    "   1\\\\\n",
    "  \\end{array} } \\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos a intentar explicar con un ejemplo gráfico que le sucede a los vectores cuando lo multiplicamos por una matriz $A$. Considera que tenemos la siguiente matriz:<br><br>\n",
    "\n",
    "<center>$A = \\left( {\\begin{array}{cc}\n",
    "   3 & 2\\\\\n",
    "   1 & 4\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "\n",
    "Si la multiplicamos por un vector:<br>\n",
    "\n",
    "<center>$A = \\left( {\\begin{array}{cc}\n",
    "   3 & 2\\\\\n",
    "   1 & 4\\\\\n",
    "  \\end{array} } \\right)\\left( {\\begin{array}{cc}\n",
    "   x\\\\\n",
    "   y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   3x + 2y\\\\\n",
    "   x + 4y\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "\n",
    "Para poder ver el efecto que hace dicha operación vamos a considerar un cuadrado y vamos a resolver dicha ecuación para cada vértice del cuadrado, es decir:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.axes()\n",
    "rectangle = plt.Rectangle((0,0), 1, 1)\n",
    "plt.gca().add_patch(rectangle)\n",
    "plt.xlim(-0.25, 1.25)\n",
    "plt.ylim(-0.25, 1.25)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Los vértices del cuadrado son:\n",
    "\n",
    "<center>$v_1 = \\left( {\\begin{array}{cc}\n",
    "   x\\\\\n",
    "   y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   0\\\\\n",
    "   0\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "<center>$v_2 = \\left( {\\begin{array}{cc}\n",
    "   x\\\\\n",
    "   y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   0\\\\\n",
    "   1\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "<center>$v_3 = \\left( {\\begin{array}{cc}\n",
    "   x\\\\\n",
    "   y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   1\\\\\n",
    "   1\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "<center>$v_4 = \\left( {\\begin{array}{cc}\n",
    "   x\\\\\n",
    "   y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   1\\\\\n",
    "   0\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "\n",
    "Si evaluamos la ecuación anterior en dichos puntos, obtenemos:<br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "A = np.array([[3, 2],\n",
    "              [1, 4]])\n",
    "\n",
    "v1 = np.array([0, 0])\n",
    "v2 = np.array([0, 1])\n",
    "v3 = np.array([1, 1])\n",
    "v4 = np.array([1, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 0]\n",
      "[2 4]\n",
      "[5 5]\n",
      "[3 1]\n"
     ]
    }
   ],
   "source": [
    "A_v1 = np.dot(A, v1)\n",
    "A_v2 = np.dot(A, v2)\n",
    "A_v3 = np.dot(A, v3)\n",
    "A_v4 = np.dot(A, v4)\n",
    "\n",
    "print(A_v1)\n",
    "print(A_v2)\n",
    "print(A_v3)\n",
    "print(A_v4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En resumen, hemos hecho estas operaciones:\n",
    "\n",
    "<center>$A(v_1) = \\left( {\\begin{array}{cc}\n",
    "   3 & 2\\\\\n",
    "   1 & 4\\\\\n",
    "  \\end{array} } \\right)\\left( {\\begin{array}{cc}\n",
    "   x\\\\\n",
    "   y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   3x + 2y\\\\\n",
    "   x + 4y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   0\\\\\n",
    "   0\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "<center>$A(v_2) = \\left( {\\begin{array}{cc}\n",
    "   3 & 2\\\\\n",
    "   1 & 4\\\\\n",
    "  \\end{array} } \\right)\\left( {\\begin{array}{cc}\n",
    "   x\\\\\n",
    "   y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   0 + 2\\\\\n",
    "   0 + 4\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   2\\\\\n",
    "   4\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "<center>$A(v_3) = \\left( {\\begin{array}{cc}\n",
    "   3 & 2\\\\\n",
    "   1 & 4\\\\\n",
    "  \\end{array} } \\right)\\left( {\\begin{array}{cc}\n",
    "   x\\\\\n",
    "   y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   3 + 2\\\\\n",
    "   1 + 4\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   5\\\\\n",
    "   5\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "<center>$A(v_4) = \\left( {\\begin{array}{cc}\n",
    "   3 & 2\\\\\n",
    "   1 & 4\\\\\n",
    "  \\end{array} } \\right)\\left( {\\begin{array}{cc}\n",
    "   x\\\\\n",
    "   y\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   3 + 0\\\\\n",
    "   1 + 0\\\\\n",
    "  \\end{array} } \\right) = \\left( {\\begin{array}{cc}\n",
    "   3\\\\\n",
    "   1\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "\n",
    "Podemos graficarlo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib.patches import Polygon\n",
    "\n",
    "pts = np.array([A_v1, A_v2, A_v3, A_v4])\n",
    "p = Polygon(pts, closed=False)\n",
    "ax = plt.gca()\n",
    "ax.add_patch(p)\n",
    "ax.set_xlim(-0.25, 5.25)\n",
    "ax.set_ylim(-0.25, 5.25)\n",
    "ax.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXwAAAD4CAYAAADvsV2wAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAAAsTAAALEwEAmpwYAAATgElEQVR4nO3df4zcdZ3H8efbVoTrIijVlaPF1lxJ7FF/0A3IaWQ34GXhj/YSOa5EEC5go3c1l0AuV8MFDf4jXvAuknraICkqsiLxpMEa7sRuyInlSg+kFFKo9U6KHFXAJisoNL7vj/lqxmW3nel8d6ZfPs9HMun3x6fzec1k97Xf+e7MfiMzkSS9+r1m0AEkSf1h4UtSISx8SSqEhS9JhbDwJakQ8wcdYDYLFy7MJUuWdDz+V7/6FQsWLJi7QDVrWl4wcz80LS80L3PT8kJ3mXfs2PGLzHzTjDsz86i8rVy5MruxdevWrsYPWtPyZpq5H5qWN7N5mZuWN7O7zMADOUuvekpHkgph4UtSIWop/Ii4OSL2R8Qjs+z/UEQ8HBE7I+K+iHhnHfNKkjpX1xH+JmD8EPt/ApyTmSuATwMba5pXktShWt6lk5n3RsSSQ+y/r211G7CojnklSZ0bxDn8K4DvDmBeSSpaZE1/LbM6wr8rM08/xJgx4AvA+zLz2Rn2rwXWAgwPD6+cmJjoeP6pqSmGhoa6jT0wTcsLZu6HpuWF5mVuWl7oLvPY2NiOzByZceds79fs9gYsAR45xP53AD8GTuvk/nwf/tHHzHOvaXkzm5e5aXkzG/Y+/Ig4FfgWcGlmPt6POSVJf6iWX9pGxG3AKLAwIvYBnwReC5CZXwSuBU4CvhARAAdztpcckqQ5Ude7dC4+zP4rgSvrmEuSdGT8pK0kFcLCl6RCWPiSVAgLX5IKYeFLUiEsfEkqhIUvSYWw8CWpEBa+JBXCwpekQlj4klQIC1+SCmHhS1IhLHxJKoSFL0mFsPAlqRAWviQVwsKXpEJY+JJUCAtfkgpRS+FHxM0RsT8iHpllf0TE5yNiT0Q8HBFn1DGvJKlzdR3hbwLGD7H/fGBZdVsL/GtN80qSOlRL4WfmvcBzhxiyGvhKtmwDToyIk+uYW5LUmX6dwz8FeLJtfV+1TZLUJ5GZ9dxRxBLgrsw8fYZ9dwGfycz/rNbvAf4hMx+YNm4trVM+DA8Pr5yYmOh4/qmpKYaGho78AfTZ/ucO8MyLg07RneHjMPMca1peaF7mpSfMa1RXQHf9NjY2tiMzR2baN7/WVLN7Cljctr6o2vYHMnMjsBFgZGQkR0dHO55gcnKSbsYP2o233skNO/v19Nfj6hUHzTzHmpYXmpd50/iCRnUF1Ndv/Tqlsxn4cPVunfcABzLz6T7NLUmipiP8iLgNGAUWRsQ+4JPAawEy84vAFuACYA/wAvDXdcwrSepcLYWfmRcfZn8Cf1vHXJKkI+MnbSWpEBa+JBXCwpekQlj4klQIC1+SCmHhS1IhLHxJKoSFL0mFsPAlqRAWviQVwsKXpEJY+JJUCAtfkgph4UtSISx8SSqEhS9JhbDwJakQFr4kFcLCl6RCWPiSVIhaCj8ixiNid0TsiYj1M+w/NSK2RsSDEfFwRFxQx7ySpM71XPgRMQ/YAJwPLAcujojl04b9I3B7Zr4bWAN8odd5JUndqeMI/0xgT2buzcyXgAlg9bQxCby+Wj4B+FkN80qSujC/hvs4BXiybX0fcNa0MZ8C/j0iPg4sAM6rYV5JUhciM3u7g4gLgfHMvLJavxQ4KzPXtY25qprrhog4G/gycHpm/nbafa0F1gIMDw+vnJiY6DjH1NQUQ0NDPT2Wftr/3AGeeXHQKbozfBxmnmNNywvNy7z0hHmN6grort/GxsZ2ZObITPvqOMJ/Cljctr6o2tbuCmAcIDN/GBHHAguB/e2DMnMjsBFgZGQkR0dHOw4xOTlJN+MH7cZb7+SGnXU8/f1z9YqDZp5jTcsLzcu8aXxBo7oC6uu3Os7hbweWRcTSiDiG1i9lN08b81PgXICIeDtwLPDzGuaWJHWo58LPzIPAOuBu4DFa78bZFRHXRcSqatjVwEci4kfAbcDl2eu5JElSV2p5HZaZW4At07Zd27b8KPDeOuaSJB0ZP2krSYWw8CWpEBa+JBXCwpekQlj4klQIC1+SCmHhS1IhLHxJKoSFL0mFsPAlqRAWviQVwsKXpEJY+JJUCAtfkgph4UtSISx8SSqEhS9JhbDwJakQFr4kFcLCl6RC1FL4ETEeEbsjYk9ErJ9lzEUR8WhE7IqIr9cxrySpc/N7vYOImAdsAD4A7AO2R8TmzHy0bcwy4BPAezPz+Yh4c6/zSpK6U8cR/pnAnszcm5kvARPA6mljPgJsyMznATJzfw3zSpK6UEfhnwI82ba+r9rW7jTgtIj4QURsi4jxGuaVJHUhMrO3O4i4EBjPzCur9UuBszJzXduYu4CXgYuARcC9wIrM/OW0+1oLrAUYHh5eOTEx0XGOqakphoaGenos/bT/uQM88+KgU3Rn+DjMPMealheal3npCfMa1RXQXb+NjY3tyMyRmfb1fA4feApY3La+qNrWbh9wf2a+DPwkIh4HlgHb2wdl5kZgI8DIyEiOjo52HGJycpJuxg/ajbfeyQ0763j6++fqFQfNPMealheal3nT+IJGdQXU1291nNLZDiyLiKURcQywBtg8bcy3gVGAiFhI6xTP3hrmliR1qOfCz8yDwDrgbuAx4PbM3BUR10XEqmrY3cCzEfEosBX4+8x8tte5JUmdq+V1WGZuAbZM23Zt23ICV1U3SdIA+ElbSSqEhS9JhbDwJakQFr4kFcLCl6RCWPiSVAgLX5IKYeFLUiEsfEkqhIUvSYWw8CWpEBa+JBXCwpekQlj4klQIC1+SCmHhS1IhLHxJKoSFL0mFsPAlqRAWviQVopbCj4jxiNgdEXsiYv0hxn0wIjIiRuqYV5LUuZ4LPyLmARuA84HlwMURsXyGcccDfwfc3+uckqTu1XGEfyawJzP3ZuZLwASweoZxnwauB35dw5ySpC7VUfinAE+2re+rtv1eRJwBLM7M79QwnyTpCMyf6wki4jXA54DLOxi7FlgLMDw8zOTkZMfzTE1NdTV+0IaPg6tXHBx0jK6Yee41LS80L3PTugLqy1xH4T8FLG5bX1Rt+53jgdOByYgAeAuwOSJWZeYD7XeUmRuBjQAjIyM5OjracYjJyUm6GT9oN956JzfsnPOft7W6esVBM8+xpuWF5mXeNL6gUV0B9fVbHad0tgPLImJpRBwDrAE2/25nZh7IzIWZuSQzlwDbgFeUvSRpbvVc+Jl5EFgH3A08Btyembsi4rqIWNXr/UuS6lHL67DM3AJsmbbt2lnGjtYxpySpO37SVpIKYeFLUiEsfEkqhIUvSYWw8CWpEBa+JBXCwpekQlj4klQIC1+SCmHhS1IhLHxJKoSFL0mFsPAlqRAWviQVwsKXpEJY+JJUCAtfkgph4UtSISx8SSqEhS9Jhail8CNiPCJ2R8SeiFg/w/6rIuLRiHg4Iu6JiLfWMa8kqXM9F35EzAM2AOcDy4GLI2L5tGEPAiOZ+Q7gDuCzvc4rSepOHUf4ZwJ7MnNvZr4ETACr2wdk5tbMfKFa3QYsqmFeSVIX6ij8U4An29b3VdtmcwXw3RrmlSR1ITKztzuIuBAYz8wrq/VLgbMyc90MYy8B1gHnZOZvZti/FlgLMDw8vHJiYqLjHFNTUwwNDR3ZgxiA/c8d4JkXB52iO8PHYeY51rS80LzMS0+Y16iugO76bWxsbEdmjsy0b34NWZ4CFretL6q2/YGIOA+4hlnKHiAzNwIbAUZGRnJ0dLTjEJOTk3QzftBuvPVObthZx9PfP1evOGjmOda0vNC8zJvGFzSqK6C+fqvjlM52YFlELI2IY4A1wOb2ARHxbuBLwKrM3F/DnJKkLvVc+Jl5kNZpmruBx4DbM3NXRFwXEauqYf8EDAHfjIiHImLzLHcnSZojtbwOy8wtwJZp265tWz6vjnkkSUfOT9pKUiEsfEkqhIUvSYWw8CWpEBa+JBXCwpekQlj4klQIC1+SCmHhS1IhLHxJKoSFL0mFsPAlqRAWviQVwsKXpEJY+JJUCAtfkgph4UtSISx8SSqEhS9JhbDwJakQtRR+RIxHxO6I2BMR62fY/7qI+Ea1//6IWFLHvJKkzvVc+BExD9gAnA8sBy6OiOXThl0BPJ+ZfwL8M3B9r/NKkrpTxxH+mcCezNybmS8BE8DqaWNWA7dUy3cA50ZE1DC3JKlDdRT+KcCTbev7qm0zjsnMg8AB4KQa5pYkdWj+oAO0i4i1wFqA4eFhJicnO/6/U1NTXY0ftKUnzGPT+IJBx+jK1NSUmedY0/JC8zI3rSugxsyZ2dMNOBu4u239E8Anpo25Gzi7Wp4P/AKIQ93vypUrsxtbt27tavygNS1vppn7oWl5M5uXuWl5M7vLDDyQs/RqHad0tgPLImJpRBwDrAE2TxuzGbisWr4Q+H4VTJLUJz2f0snMgxGxjtZR/Dzg5szcFRHX0fpJsxn4MvDViNgDPEfrh4IkqY9qOYefmVuALdO2Xdu2/GvgL+uYS5J0ZPykrSQVwsKXpEJY+JJUCAtfkgph4UtSISx8SSqEhS9JhbDwJakQFr4kFcLCl6RCWPiSVAgLX5IKYeFLUiEsfEkqhIUvSYWw8CWpEBa+JBXCwpekQlj4klQIC1+SCtFT4UfEGyPiPyLiierfN8ww5l0R8cOI2BURD0fEX/UypyTpyPR6hL8euCczlwH3VOvTvQB8ODP/FBgH/iUiTuxxXklSl3ot/NXALdXyLcBfTB+QmY9n5hPV8s+A/cCbepxXktSlXgt/ODOfrpb/Dxg+1OCIOBM4Bvhxj/NKkroUmXnoARHfA94yw65rgFsy88S2sc9n5ivO41f7TgYmgcsyc9ssY9YCawGGh4dXTkxMdPAQWqamphgaGup4/KA1LS+YuR+alheal7lpeaG7zGNjYzsyc2TGnZl5xDdgN3BytXwysHuWca8H/hu4sNP7XrlyZXZj69atXY0ftKblzTRzPzQtb2bzMjctb2Z3mYEHcpZe7fWUzmbgsmr5MuDO6QMi4hjg34CvZOYdPc4nSTpCvRb+Z4APRMQTwHnVOhExEhE3VWMuAt4PXB4RD1W3d/U4rySpS/N7+c+Z+Sxw7gzbHwCurJa/Bnytl3kkSb3zk7aSVAgLX5IKcdi3ZQ5KRPwc+N8u/stC4BdzFGcuNC0vmLkfmpYXmpe5aXmhu8xvzcwZP9x61BZ+tyLigZztvadHoablBTP3Q9PyQvMyNy0v1JfZUzqSVAgLX5IK8Woq/I2DDtClpuUFM/dD0/JC8zI3LS/UlPlVcw5fknRor6YjfEnSIVj4klSIxhZ+Uy6vGBHjEbE7IvZExCuuCBYRr4uIb1T774+IJf3OOEOmw2W+KiIerZ7TeyLirYPI2ZbnkHnbxn0wIjIiBv6WvE4yR8RF1fO8KyK+3u+MM+Q53NfFqRGxNSIerL42LhhEzrY8N0fE/oh4ZJb9ERGfrx7PwxFxRr8zTstzuLwfqnLujIj7IuKdXU8y25/RPNpvwGeB9dXyeuD6GcacBiyrlv8YeBo4sY8Z59G62MvbaF345UfA8mlj/gb4YrW8BvjGgJ/XTjKPAX9ULX9skJk7yVuNOx64F9gGjDTgOV4GPAi8oVp/cwMybwQ+Vi0vB/5nwJnfD5wBPDLL/guA7wIBvAe4/yjP+2dtXw/nH0nexh7h04zLK54J7MnMvZn5EjBBK3e79sdxB3BuREQfM0532MyZuTUzX6hWtwGL+pyxXSfPMcCngeuBX/cz3Cw6yfwRYENmPg+Qmfv7nHG6TjInrWtfAJwA/KyP+V4hM+8FnjvEkNW0/mx7ZuuiTCdWF2oaiMPlzcz7fvf1wBF+3zW58JtwecVTgCfb1vdV22Yck5kHgQPASX1JN7NOMre7gtZR0qAcNm/1Un1xZn6nn8EOoZPn+DTgtIj4QURsi4jxvqWbWSeZPwVcEhH7gC3Ax/sT7Yh1+7V+NDmi77ue/jzyXDvM5RV/LzMzImZ9f2n1U/urtC6v+Nt6U5YrIi4BRoBzBp1lNhHxGuBzwOUDjtKt+bRO64zSOpK7NyJWZOYvBxnqMC4GNmXmDRFxNvDViDjd77l6RcQYrcJ/X7f/96gu/Mw8b7Z9EfFMRJycmU9XhT7jS96IeD3wHeCanOVaunPoKWBx2/qiattMY/ZFxHxaL4Wf7U+8GXWSmYg4j9YP3nMy8zd9yjaTw+U9HjgdmKzOlL0F2BwRq7J13YZB6OQ53kfrHO3LwE8i4nFaPwC29yfiK3SS+QpgHCAzfxgRx9L6o1+DPh01m46+1o8mEfEO4Cbg/Gxdj6QrTT6l04TLK24HlkXE0irLGlq527U/jguB72f1W5kBOWzmiHg38CVg1VFwbvmQeTPzQGYuzMwlmbmE1rnPQZY9dPZ18W1aR/dExEJap3j29jHjdJ1k/inVBZEi4u3AscDP+5qyO5uBD1fv1nkPcKDtNPFRJyJOBb4FXJqZjx/RnQzyt9I9/kb7JOAe4Ange8Abq+0jwE3V8iXAy8BDbbd39TnnBcDjtH53cE217TpapQOtb4pvAnuA/wLedhQ8t4fL/D3gmbbndPPRnHfa2EkG/C6dDp/joHUq6lFgJ7CmAZmXAz+g9Q6eh4A/H3De22i9M+9lWq+YrgA+Cny07TneUD2enYP+uugg703A823fd7NerHy2m39aQZIK0eRTOpKkLlj4klQIC1+SCmHhS1IhLHxJKoSFL0mFsPAlqRD/D4Cni8aAopoAAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.axes()\n",
    "rectangle = plt.Rectangle((0,0), 1, 1)\n",
    "plt.gca().add_patch(rectangle)\n",
    "plt.xlim(-0.25, 1.25)\n",
    "plt.ylim(-0.25, 1.25)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si recordamos bien, una de las componentes de los vectores (cuando pensamos en ellos geométricamente) es la `dirección`. **¿Se mantiene aquí la dirección de los vectores?**... Vamos a verlo...😄\n",
    "\n",
    "- El punto `v2` ha pasado de $(0, 1)$ a $(2, 4)$\n",
    "- El punto `v4` ha pasado de $(1, 0)$ a $(3, 1)$\n",
    "- El punto `v3` ha pasado de $(1, 1)$ a $(5, 5)$\n",
    "\n",
    "Vemos como el punto `v3` es el único que ha mantenido su dirección, y de hecho se ha dilatado por 5. Este vector que ha mantenido su dirección se denomina **autovector** y el factor de dilatación es el **autovalor** correspondiente.\n",
    "\n",
    "<div class=\"alert alert-success\">\n",
    "Suponiendo que tenemos una matriz cuadrada $A$ de $nxn$, ¿existe algún vector $x$ distinto de $0$ para el cual $Ax$ es un escalar múltiplo de $x$? Si escribimos esta pregunta en lenguaje matemático, nos encontramos con la definición de los autovalores y autovectores:<br><br>\n",
    "    \n",
    "<center>$Ax = \\lambda x$</center>\n",
    "</div>\n",
    "\n",
    "Podemos desarrollarla un poco:\n",
    "\n",
    "<center>$Ax = \\lambda x$</center>\n",
    "\n",
    "Moviendo un miembro al otro lado:\n",
    "\n",
    "<center>$Ax - \\lambda x = 0$</center>\n",
    "\n",
    "Multiplicando por $I$:\n",
    "<center>$(A - \\lambda I)x= 0$</center>\n",
    "\n",
    "Al término $(A - \\lambda I)$ se denomina **matriz de coeficientes**. Queremos obtener los resultados para cuando esta ecuación sea distinto de cero:<br><br>\n",
    "\n",
    "<center>$(A - \\lambda I)x= 0$</center>\n",
    "\n",
    "Un escalar $\\lambda$ es un valor propio de una matriz $A$ de $(nxn)$ si, y sólo si, $\\lambda$ satisface la ecuación característica:<br><br>\n",
    "\n",
    "<center>$det(A - \\lambda I) = 0$</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-success\">\n",
    "Considera la siguiente matriz<br><br>\n",
    "    \n",
    "<center>$A = \\left( {\\begin{array}{cc}\n",
    "   3 & 2\\\\\n",
    "   7 & -2\\\\\n",
    "  \\end{array} } \\right)\n",
    "$</center><br>\n",
    "</div>\n",
    "\n",
    "Podemos encontrar los autovalores de la siguiente forma:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Autovalores:  [ 5. -4.]\n",
      "Autovectores: \n",
      " [[ 0.70710678 -0.27472113]\n",
      " [ 0.70710678  0.96152395]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "A = np.array([[3, 2],\n",
    "              [7, -2]])\n",
    "\n",
    "x, v = np.linalg.eig(A)\n",
    "print(\"Autovalores: \", x)\n",
    "print(\"Autovectores: \\n\", v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O usando `sympy`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Autovalores  {5: 1, -4: 1}\n",
      "Autovectores  [(-4, 1, [Matrix([\n",
      "[-2/7],\n",
      "[   1]])]), (5, 1, [Matrix([\n",
      "[1],\n",
      "[1]])])]\n"
     ]
    }
   ],
   "source": [
    "import sympy\n",
    "\n",
    "A = sympy.Matrix([[3, 2],\n",
    "                  [7, -2]])\n",
    "\n",
    "print(\"Autovalores \", A.eigenvals())\n",
    "print(\"Autovectores \", A.eigenvects())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9.1-  Eigenvalues y Eigenvectors en ML\n",
    "\n",
    "El cálculo de autovectores y autovalores tiene muchísimas aplicaciones, como puede ser el cálculo de componentes principales en Machine Learning.\n",
    "\n",
    "Veamos un ejemplo sencillo:  es la de obtener los ejes principales de una matriz de dispersión (o de covarianza) de un conjunto de muestras. En otras palabras, ver en qué direcciones tienden a estar alineados un montón de puntos.\n",
    "\n",
    "Un ejemplo en dos dimensiones: si calculamos la matriz de covarianza de los puntos de la figura, sus dos autovectores nos indican las direcciones marcadas por flechas, cuya longitud está dada por los autovalores. Y el autovector del mayor autovalor nos dice la dirección principal en la que se alinean los puntos.  \n",
    "\n",
    "<img src=\"Images/PCA.png\" width=40%/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9.2-  Singular Value Decomposition - SVD\n",
    "\n",
    "Dada una matrix $X$ con tamaño $(m,n)$, puede ser expresada como:\n",
    "\n",
    "<img src=\"Images/svd.png\" width=30%/>\n",
    "\n",
    "Donde:\n",
    "\n",
    "- r es el rango de la matrix X\n",
    "- U es una matriz $(m ,r)$ y orthornormal\n",
    "- V es una matriz $(n, r)$ y orthonormal\n",
    "- $\\Sigma$ es una matriz diagonal de $(r, r)$ - cuyos valores se llaman _singular values_\n",
    "\n",
    "1- Podemos usar `SVD` de una forma cualitativa, es decir, que nos sirve para extraer información de nuestros datos.\n",
    "2- O de una forma cuantitativa, que nos permite reducir la dimensionalidad de nuestra matrix $X$ y seguir manteniendo la información importante.\n",
    "\n",
    "<img src=\"Images/svd_movies.png\" width=60%/>\n",
    "\n",
    "Donde:\n",
    "\n",
    "- $U$ es una matriz que relaciona: user-concept\n",
    "- $V$ es una matriz que relaciona: movie-concept\n",
    "- $\\Sigma$: importancia de cada concepto\n",
    "\n",
    "Usando Python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "#user vs movie, a 7x5 matrix\n",
    "A = np.array([[1, 1, 1, 0, 0],\\\n",
    "              [3, 3, 3, 0, 0],\\\n",
    "              [4, 4, 4, 0, 0],\\\n",
    "              [5, 5, 5, 0, 0],\\\n",
    "              [0, 2, 0, 4, 4],\\\n",
    "              [0, 0, 0, 5, 5],\\\n",
    "              [0, 1, 0, 2, 2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U : \n",
      " [[-0.14 -0.02 -0.01  0.56 -0.38]\n",
      " [-0.41 -0.07 -0.03  0.21  0.76]\n",
      " [-0.55 -0.09 -0.04 -0.72 -0.18]\n",
      " [-0.69 -0.12 -0.05  0.34 -0.23]\n",
      " [-0.15  0.59  0.65  0.    0.2 ]\n",
      " [-0.07  0.73 -0.68  0.    0.  ]\n",
      " [-0.08  0.3   0.33  0.   -0.4 ]]\n",
      "s : \n",
      " [[12.48  0.    0.    0.    0.  ]\n",
      " [ 0.    9.51  0.    0.    0.  ]\n",
      " [ 0.    0.    1.35  0.    0.  ]\n",
      " [ 0.    0.    0.    0.    0.  ]\n",
      " [ 0.    0.    0.    0.    0.  ]]\n",
      "Vt : \n",
      " [[-0.56 -0.59 -0.56 -0.09 -0.09]\n",
      " [-0.13  0.03 -0.13  0.7   0.7 ]\n",
      " [-0.41  0.8  -0.41 -0.09 -0.09]\n",
      " [-0.71  0.    0.71  0.    0.  ]\n",
      " [ 0.   -0.    0.   -0.71  0.71]]\n"
     ]
    }
   ],
   "source": [
    "U, s, Vt = np.linalg.svd(A, full_matrices=False)\n",
    "S = np.diag(s)\n",
    "\n",
    "print(\"U : \\n\", np.round(U, 2))\n",
    "print(\"s : \\n\", np.round(S, 2))\n",
    "print(\"Vt : \\n\", np.round(Vt, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  1.  1. -0. -0.]\n",
      " [ 3.  3.  3. -0. -0.]\n",
      " [ 4.  4.  4. -0. -0.]\n",
      " [ 5.  5.  5. -0. -0.]\n",
      " [-0.  2. -0.  4.  4.]\n",
      " [-0.  0. -0.  5.  5.]\n",
      " [-0.  1. -0.  2.  2.]]\n"
     ]
    }
   ],
   "source": [
    "A_reconstruida = U.dot(S.dot(Vt)) # np.dot(U, np.dot(S, Vt))\n",
    "print(np.round(A_reconstruida, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 1, 1, 0, 0],\n",
       "       [3, 3, 3, 0, 0],\n",
       "       [4, 4, 4, 0, 0],\n",
       "       [5, 5, 5, 0, 0],\n",
       "       [0, 2, 0, 4, 4],\n",
       "       [0, 0, 0, 5, 5],\n",
       "       [0, 1, 0, 2, 2]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(7, 5)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(7, 5)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5, 5)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5, 5)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vt.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Otra utilidad que nos proporciona este tipo de descomposiciones, es que podemos aplicar una reducción de dimensionalidad a la matriz original y mantener la información relevante.\n",
    "\n",
    "Para entender este concepto, vamos a usar esta página web: http://timbaumann.info/svd-image-compression-demo/\n",
    "\n",
    "Ahora vamos a hacerlo en Python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "El maximo valor de r es:  5\n",
      "1\n",
      "Numero de components:  1\n",
      "\n",
      " A reconstruida: \n",
      " [[0.97 1.02 0.97 0.15 0.15]\n",
      " [2.9  3.05 2.9  0.46 0.46]\n",
      " [3.86 4.07 3.86 0.62 0.62]\n",
      " [4.83 5.09 4.83 0.77 0.77]\n",
      " [1.07 1.13 1.07 0.17 0.17]\n",
      " [0.51 0.53 0.51 0.08 0.08]\n",
      " [0.54 0.57 0.54 0.09 0.09]]\n",
      "\n",
      " A: \n",
      " [[1 1 1 0 0]\n",
      " [3 3 3 0 0]\n",
      " [4 4 4 0 0]\n",
      " [5 5 5 0 0]\n",
      " [0 2 0 4 4]\n",
      " [0 0 0 5 5]\n",
      " [0 1 0 2 2]]\n",
      "\n",
      " SSE:  92.22427221863023\n",
      "\n",
      " Comp ratio:  0.37142857142857144\n",
      "\n",
      "-------------------------------------------------\n",
      "2\n",
      "Numero de components:  2\n",
      "\n",
      " A reconstruida: \n",
      " [[ 0.99  1.01  0.99 -0.   -0.  ]\n",
      " [ 2.98  3.04  2.98 -0.   -0.  ]\n",
      " [ 3.98  4.05  3.98 -0.01 -0.01]\n",
      " [ 4.97  5.06  4.97 -0.01 -0.01]\n",
      " [ 0.36  1.29  0.36  4.08  4.08]\n",
      " [-0.37  0.73 -0.37  4.92  4.92]\n",
      " [ 0.18  0.65  0.18  2.04  2.04]]\n",
      "\n",
      " A: \n",
      " [[1 1 1 0 0]\n",
      " [3 3 3 0 0]\n",
      " [4 4 4 0 0]\n",
      " [5 5 5 0 0]\n",
      " [0 2 0 4 4]\n",
      " [0 0 0 5 5]\n",
      " [0 1 0 2 2]]\n",
      "\n",
      " SSE:  1.810530940559784\n",
      "\n",
      " Comp ratio:  0.7428571428571429\n",
      "\n",
      "-------------------------------------------------\n",
      "3\n",
      "Numero de components:  3\n",
      "\n",
      " A reconstruida: \n",
      " [[ 1.  1.  1. -0. -0.]\n",
      " [ 3.  3.  3. -0. -0.]\n",
      " [ 4.  4.  4. -0. -0.]\n",
      " [ 5.  5.  5. -0. -0.]\n",
      " [ 0.  2. -0.  4.  4.]\n",
      " [-0.  0. -0.  5.  5.]\n",
      " [ 0.  1. -0.  2.  2.]]\n",
      "\n",
      " A: \n",
      " [[1 1 1 0 0]\n",
      " [3 3 3 0 0]\n",
      " [4 4 4 0 0]\n",
      " [5 5 5 0 0]\n",
      " [0 2 0 4 4]\n",
      " [0 0 0 5 5]\n",
      " [0 1 0 2 2]]\n",
      "\n",
      " SSE:  8.661231396350227e-29\n",
      "\n",
      " Comp ratio:  1.1142857142857143\n",
      "\n",
      "-------------------------------------------------\n",
      "4\n",
      "Numero de components:  4\n",
      "\n",
      " A reconstruida: \n",
      " [[ 1.  1.  1. -0. -0.]\n",
      " [ 3.  3.  3. -0. -0.]\n",
      " [ 4.  4.  4. -0. -0.]\n",
      " [ 5.  5.  5. -0. -0.]\n",
      " [-0.  2. -0.  4.  4.]\n",
      " [-0.  0. -0.  5.  5.]\n",
      " [-0.  1. -0.  2.  2.]]\n",
      "\n",
      " A: \n",
      " [[1 1 1 0 0]\n",
      " [3 3 3 0 0]\n",
      " [4 4 4 0 0]\n",
      " [5 5 5 0 0]\n",
      " [0 2 0 4 4]\n",
      " [0 0 0 5 5]\n",
      " [0 1 0 2 2]]\n",
      "\n",
      " SSE:  6.356969121772307e-29\n",
      "\n",
      " Comp ratio:  1.4857142857142858\n",
      "\n",
      "-------------------------------------------------\n",
      "5\n",
      "Numero de components:  5\n",
      "\n",
      " A reconstruida: \n",
      " [[ 1.  1.  1. -0. -0.]\n",
      " [ 3.  3.  3. -0. -0.]\n",
      " [ 4.  4.  4. -0. -0.]\n",
      " [ 5.  5.  5. -0. -0.]\n",
      " [-0.  2. -0.  4.  4.]\n",
      " [-0.  0. -0.  5.  5.]\n",
      " [-0.  1. -0.  2.  2.]]\n",
      "\n",
      " A: \n",
      " [[1 1 1 0 0]\n",
      " [3 3 3 0 0]\n",
      " [4 4 4 0 0]\n",
      " [5 5 5 0 0]\n",
      " [0 2 0 4 4]\n",
      " [0 0 0 5 5]\n",
      " [0 1 0 2 2]]\n",
      "\n",
      " SSE:  6.356969121772307e-29\n",
      "\n",
      " Comp ratio:  1.8571428571428572\n",
      "\n",
      "-------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "U, s, Vt = np.linalg.svd(A, full_matrices=False)\n",
    "S = np.diag(s)\n",
    "r = S.shape[0]\n",
    "\n",
    "print(\"El maximo valor de r es: \", r)\n",
    "# U - (m x r)\n",
    "for n_componentes in range(1, r+1):\n",
    "    print(n_componentes)\n",
    "    U_reducida = U[:, :n_componentes] # m x r\n",
    "    S_reducida = S[:n_componentes, :n_componentes] # r x r\n",
    "    Vt_reducida = Vt[:n_componentes, :] # r x n\n",
    "    \n",
    "    # Reconstruimos matriz\n",
    "    A_hat_reconstructed = np.dot(U_reducida, np.dot(S_reducida, Vt_reducida))\n",
    "    \n",
    "    # Error de reconstruccion\n",
    "    SSE = np.sum((A - A_hat_reconstructed)**2)\n",
    "    \n",
    "    # Ratio de compresión m × r + r + r × n\n",
    "    comp_ratio = (A.shape[1]*n_componentes + n_componentes + n_componentes*A.shape[0]) / (A.shape[1] * A.shape[0])\n",
    "    \n",
    "    print(\"Numero de components: \", n_componentes)\n",
    "    print(\"\\n A reconstruida: \\n\", np.round(A_hat_reconstructed, 2))\n",
    "    print(\"\\n A: \\n\", A)\n",
    "    print(\"\\n SSE: \", SSE)\n",
    "    print(\"\\n Comp ratio: \", comp_ratio)\n",
    "    print('\\n-------------------------------------------------')\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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