Eigenvectors And Eigenvalues

eigen.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Imports\n",
    "\n",
    "The first thing we'll need is numpy, probably the most OP machine learning library! We'll also use matplotlib too to visualise what we are doing. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib\n",
    "from matplotlib.patches import Polygon\n",
    "from matplotlib.collections import PatchCollection\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating Eigenvalues and Eigenvectors\n",
    "\n",
    "Here we have a transformation matrix. We can compute the eigenvectors and eigenvalues using numpy. Note that numpy will scale the eigenvectors to unit vectors automatically. \n",
    "\n",
    "The eigenvectors tell us which direction the transformation is being steched, and the values tell us what scale factor by which it is being stretched."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eigenvalues:\n",
      " [ 1.53027756  1.16972244] \n",
      "eigenvectors:\n",
      " [[ 0.95709203 -0.28978415]\n",
      " [ 0.28978415  0.95709203]]\n"
     ]
    }
   ],
   "source": [
    "#transformation_matrix = np.array([[0.8, 0.2], [0.55, 1.3]])\n",
    "transformation_matrix = np.array([[1.5, 0.1], [0.1, 1.2]])\n",
    "eigenvalues, eigenvectors = np.linalg.eig(transformation_matrix)\n",
    "\n",
    "print(\"eigenvalues:\\n\", eigenvalues, \"\\neigenvectors:\\n\", eigenvectors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Eigendecomposition\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\boldsymbol{M} = \\boldsymbol{Q} \\boldsymbol{\\Lambda} \\boldsymbol{Q}^{-1}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Here in the below Python code, __M__ is the result, __Q__ are the eigenvectors (note the last one is inverse) and __V__ (upside down V == lambda) is the diagonal matrix of eigenvalues. We can assert it all worked correctly like below!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "successful eigendecomposition!\n"
     ]
    }
   ],
   "source": [
    "diag = np.diag(eigenvalues)\n",
    "inverse = np.linalg.inv(eigenvectors)\n",
    "result = eigenvectors.dot(diag).dot(inverse)\n",
    "\n",
    "if np.allclose(transformation_matrix, result):\n",
    "    print(\"successful eigendecomposition!\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A Quad's Coordinates\n",
    "\n",
    "Here we define a square that will ultimately be warped by our transformation matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "original_matrix = np.array([\n",
    "    [-1, -1], \n",
    "    [-1,  1],\n",
    "    [ 1,  1],\n",
    "    [ 1, -1]\n",
    "])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting The Quad\n",
    "\n",
    "A sanity check that we have a square. Good."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fac3c3c6898>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots()\n",
    "original_quad = Polygon(original_matrix, True)\n",
    "patch = PatchCollection([original_quad], \n",
    "                        cmap=matplotlib.cm.jet,\n",
    "                        alpha=0.2)\n",
    "axes.add_collection(patch)\n",
    "_ = axes.set_xlim([-2, 2])\n",
    "_ = axes.set_ylim([-2, 2])\n",
    "plt.gca().set_aspect('equal', adjustable='box')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transforming The Quad\n",
    "\n",
    "Now we transform the quad and get a new quad."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-1.6, -1.3],\n",
       "       [-1.4,  1.1],\n",
       "       [ 1.6,  1.3],\n",
       "       [ 1.4, -1.1]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "transformed_matrix = np.dot(original_matrix,\n",
    "                            transformation_matrix)\n",
    "transformed_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting The Quad And The Transformation\n",
    "\n",
    "We can see the two quads overlaid over one another."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fac3a0760f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots()\n",
    "original_quad = Polygon(original_matrix, True)\n",
    "new_quad = Polygon(transformed_matrix, True)\n",
    "patch = PatchCollection([original_quad, new_quad], \n",
    "                        cmap=matplotlib.cm.jet,\n",
    "                        alpha=0.2)\n",
    "axes.add_collection(patch)\n",
    "_ = axes.set_xlim([-2, 2])\n",
    "_ = axes.set_ylim([-2, 2])\n",
    "plt.gca().set_aspect('equal', adjustable='box')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Studying The Direction Of The Eigenvectors\n",
    "\n",
    "Geometrically, an eigenvector points in the direction that is stretched by the transformation. Here we get these directions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.302775637732 -3.30277563773\n"
     ]
    }
   ],
   "source": [
    "xs = eigenvectors[0]\n",
    "ys = eigenvectors[1]\n",
    "\n",
    "grad_1 = ys[0] / xs[0]\n",
    "grad_2 = ys[1] / xs[1]\n",
    "\n",
    "print(grad_1, grad_2)\n",
    "\n",
    "line_x = np.linspace(-2, 2, 500)\n",
    "grad_line_1 = line_x * grad_1\n",
    "grad_line_2 = line_x * grad_2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting The Eigenvector's Directions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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no+9ssXUeF68jvJJZ+/8fjUdepiO84j0/37Czi01vH+HEwAgCTJs4jlsvaWbRpBrCWViW\nXUhcnEMB+dtS8B7guKrOFpE7gf8FfDTTvrPJahQX7+jEqxgpriTcuSFhogCYUjeO1ZNqWNhcQ3UO\nd7Tyu0IeUaSzpeBtwN/EH/8YeFBERFWdOTqtRnFxVJXDfTE66q8j3LERLo2BnP1XceX8RlZ6E57v\nFHKNIp0tBc+0id/evweYCBw998282lLQahTpi8aUvUf62Ha4lx3tPRztG2bywmv4UMcvGd+zgxPj\nF3sdom8V8ogiq7zaUtDWeZxfJBpja1sP29p6aO3oZXAkRnFImNNUzYq5jfQ3tqB7/5b6Y1ssUWSg\nYGsUpLGl4Jg2h0SkGKhltKjpDBtRnJ+I8O+vHaQ4FOLSlvEsbqlhblM1ZcW/nRn5zE3PMlDR7GGU\n/lfII4qUWwoCjwN/DLwE3A4871R9wtZ5pFQUEv7yxnlMrCxNOivSkkTmCrZGkeaWgt8Gviciexjd\nUvDOTPvNpiBP3x4aibKr8yTbD/dyvH+Y/7ZidtK2QZr45JVCHlGks6XgIHBHNvrKhaAlihP9w2cW\nWr3d1Uc0plSUFLGguYZINObseXKhc3WdBzhYzPRCkOZQvH7gON97+V0A6qtKuX52PYsn1TCjvsrZ\nYW9QuLrOAyxRAMGaQzGzvpIPXtrM4km1NFaXOfsXLIhcTtSWKCicKx4nB0fYcbiXiVVlzG6sSthm\n/LhSVuVhExtUR+9JZ9Lman0CLFEA/l3noap09A6yra2X7e09HDjWjwLXzJyYNFHkw/T9P2L23kdY\nv/LnIIV3Y5lccbk2ZIkCf9Yo1rd28tLeY3SfGgZgyoQKVi8Ks2hSDZPG52+zmkRGSqqp7tvLxO43\nODbxSk9j8RMbUTjOjzWKgeEo4ZpybpzfxMJJNdRWuLPQqrPxemJSTLhzoyWKC2A1Cse5WKM4cnKI\n+qrSpMXGD102Kc8RpS9SUs3RiVfR3LGB7Qs/53U4vuHyiMLdk6I8cmGdRyy+0OrxNw/z90+18ndP\ntXLoxIDXYV209vBKqvv2Utm33+tQfMNqFI7zakQxEo2xo713dKFVey+nhqMUhYRZDZVcN7ueCRX+\nveV8R3gll237O5o7N7Kn6m6vw/EFl0cUgU8UXq7ziESVR17aT3l8VuTiSbXMD1cXxC3o+8e10FM9\nl3DHRvbMutvrcHzBahQO83L6dkVpEX950zzCNeVO/5JcrNb59xELuVNkdZ2NKByWi0QxOBJl66ET\nbN7XzWAkxudvnpe0bYvHlzJzqb35Rq9D8A2X13mAJYqszaHoPjXMq/u72byvmzcOnWA4EqOipIil\nM+rOu0+mMeD2Og+wRJGVORRPbWvnGxvfAaCxuoybFzaxdHodi1tqnf8FMG5w/dQz8IkiG1c8Fk+q\n5Q+vnsay6XVMmzjORg/mgrlcnwBLFCnXefQOjLDl3eNMmziOWQ2J109MqRvHR+vG5SI8ExAuz6EA\nSxQJaxSHjvezeV83m/d309reS0zhI5e3JE0UxmTKRhSOG1ujeHTzATbu6uJwzyAAM+oruWPJFJZN\nr2OWh6sx/a506DjTD/yYtuabOFU13etwnFTQNQoRqQN+CEwH9gO/p6rHE7SLAm/Fnx5Q1bWZ9JtN\nY2sU7T0DhGvLWXvZJK6aUUdjtW13lw2h2DCLWv8ZUHbPudfrcJxU6COKLwLPqer9IvLF+PMvJGg3\noKq/k2FfGevoGSRce/bBP3adx2dvnGuFyBwYrGjieO0imjs2WKJIwvUaRabR3QY8HH/8MPDhDN8v\nq2Ix5TcHjvPAMztZ/b83cfXfP0fbOQutxo4oLEnkTnt4JROOb6V0yKntXJxR6COKJlVtjz/uAJLd\nY61cRLYAEeB+Vf1ZsjfMdEvBwZEom3Yf4bnWLp7b2cXRviGKQsJV0yfw17cuYNyYdRS2n0f+dIRX\nsHDXg4Q7f8WBqR/xOhzn+L5GISLrgXCCH3157BNVVRFJdq1xmqq2ichM4HkReUtV30nUMNMtBY/2\nDXHv916juryY5XMbuGlhE8vnNjB+3HtXYgbtNv1e6qlZQH95mOaODZYoEvD9iEJVk07YF5FOEWlW\n1XYRaQa6krxHW/z7XhHZCFwOJEwUmZo8YRyP/em1XDo59axISxR5JEJHeAVTD/6cUHSIWJFtJnSa\n6+s8IPMaxemtAol///m5DURkgoiUxR/XA9cBOy60o6FIlI27uvgfP9vGV39x/n9+5bQJaU2d9uO9\nMv2sPbyKntoFlA8e8ToUp/hhmn+mNYr7gR+JyD3Au8DvAYjIEuBTqvpJYAHwbyISYzQx3a+qaSWK\nSEz58WuHWL+jkxfePsKp4SgVJUWsuSQ7e1z68V6ZftbVeB1djdd5HYZzXK9PQIaJQlWPAasSvL4F\n+GT88a+BSy7m/Vvbe/n8v79JU00ZH768hRsXNHHNrIlZu7GLi/fKNMHjen0CHJ+Z2Vhdxn/82fUs\nmlSTk3M4v+7nYQqL63MowPFE0VRTzuKW2py9v9UojAv8MKJwP5XlkNUojAv8UKMIdKKwGoVxgY0o\nHOfCfh5BVRTpp6K/zeswnOCHGoX7EeaQjSi8s/yFu/idrV/1Ogwn2IjCYbbOw1tHGq6l4egrFEVO\neR2K56xG4TCbvu2t9vAKimLDNB55yetQPGcjCodZovDWsborGC6pobljg9eheMoP6zwgwInC5lB4\nS0MldDa+j3Dnr0CjXofjGT+s84AAJwqbQ+G99vBKyoa7qTv+VurGBcoP9QkIcKKwKx7e62q8npgU\n09j1gteheMYP9QlwfAp3Ltk6D++NlNTw/PLHOFk90+tQPOOHORQQ4ERhNQo3nKyZ43UInvLLiMIf\n6SwHrEZhXGA1CsdZjcK4wEYUjrN1HsYFfqlR+CPKHLARhXFBIEYUInKHiGwXkVj8PpnJ2q0WkV0i\nsie+o5inbJ2HcUVQahTbgI8Am5I1EJEi4OvALcBC4C4RWZhhvxmx6dvuaWl7iuWbPorERrwOJa8C\nMaJQ1VZV3ZWi2VJgj6ruVdVh4AeMbkXoGUsU7omFSqg78RYTu3/jdSh545d1HpCfeRQtwMExzw8B\ny5I1znRLwXR4PYdidmOVp/27KDThJmKvl7Gg90XaF97gdTh54Zd1HpDGiEJE1ovItgRfORkVqOo6\nVV2iqksaGhpy0YXNoXBQrKSSU83XUn3gWdBgjPj8Up+ADLcUTFMbMGXM88nx1zxjVzzc1DvtRlr+\n88uUndjD0ITCn7Hpl/oE5Ofy6KvAHBGZISKlwJ2MbkXoGVvn4abeqaN/k6oPrPc4kvzwyxwKyPzy\n6O+KyCHgGuAJEXkm/vokEXkSQFUjwH3AM0Ar8CNV3Z5Z2JnxukZhEotUNjMwcTE1B571OpS88NOI\nItMtBX8K/DTB64eBNWOePwk8mUlf2WQ1CncdufRTSHTI6zByLhTyVzEzkKtHrUbhrp5Za70OIW2h\nEBSHQhSFICRCcSh05rVQCIoSvFYcChES/1wWPS2QicLWeZjTgnSwZyKQicJGFIXFDvbcC1yisHUe\nbrKD3W3BSxQ2msgZO9gLlyUKcxY72E0igUsUQZhDURAH+9BJ2P5TmHYdTJzldTSBF7hE4Zc5FAVx\nsGdiZAAe/wys+BKs+ILX0QRe4BJFPq94BP5gz0RVI0xeArufskThgMAligtd52EHu4fm3QLPfRV6\n26Gm2etoAi1wiaKkSKitKLGD3Q/mrRlNFLufhiUf9zqaQAtcomisLvc6BJOuhvkwfhrsesoShcf8\nsyrFBI/I6Khi369g+JTX0QSaJQrjtnmroXYynDiYuq3JmcCdehifmbEc/uw1r6MIPBtRGLdZMdkJ\nliiMMSlZojDGpJSvLQX3i8hbIvKGiGzJpE9jTP5lWsw8vaXgv6XRdqWqHs2wP2OMBzK9uW4r2OxF\nkyexKPR3Q1VuNoYyyeWrRqHAL0XktfiWgcZcuO/eCj/5pNdRBFLKEYWIrAfCCX70ZVX9eZr9XK+q\nbSLSCDwrIjtVNeEO6PnYe9T41OSr4OVvwGAPlNd6HU2gpBxRqOqNqro4wVe6SQJVbYt/72J0H5Cl\n52mb871HjU/NWwOxCOx5zutIAifnpx4iUiki1acfAzczWgQ15sJMWQoVdaOLxExe5XxLQaAJeFFE\n3gQ2A0+o6tOZ9GsCKlQEcz8Ab/8SohGvowmUnG8pqKp7gcsy6ceYM+auhjcfhYOvwPTrvI4mMGxm\npvGX2augqHT0Fnkmb2z1qPGXsmr4419A+BKvIwkUSxTGf6Ze7XUEgWOnHsaYlCxRGGNSskRhjEnJ\nEoUxJiVLFMaYlCxRGP/asx4e+TBEhr2OpOBZojD+FY3A3g3w7oteR1LwLFEY/5q5HIorbJFYHlii\nMP5VUgGzVsKup0Hzt0t9EFmiMP42dzX0HIDO7V5HUtAsURh/m7t69LudfuSUJQrjb9VN0HKlrSbN\nMVsUZvxv6b1wsmO0TmF3hM8JSxTG/y670+sICp6dehhjUrJEYYxJKdOb6z4gIjtFZKuI/FRExidp\nt1pEdonIHhH5YiZ9GmPyL9MRxbPAYlW9FNgNfOncBiJSBHwduAVYCNwlIgsz7NcYk0cZJQpV/aWq\nnr5v+svA5ATNlgJ7VHWvqg4DPwBuy6RfY0x+ZfOqxyeAHyZ4vQU4OOb5IWBZsjcZu6UgMCQihbhZ\nUD1QiDu7F+rngsL9bPPSaZSVvUdF5MtABPj+hUSYiKquA9bF33eLqi7J9D1dY5/Lfwr1s4nIlnTa\npUwUqnpjio7uBj4IrFJNuDKnDZgy5vnk+GvGGJ/I9KrHauCvgLWq2p+k2avAHBGZISKlwJ3A45n0\na4zJr0yvejwIVAPPisgbIvJNOHvv0Xix8z7gGaAV+JGqprvUb12G8bnKPpf/FOpnS+tzSeKzBWOM\n+S2bmWmMSckShTEmJacTRbpTxP1IRO4Qke0iEhMR3192K9Rp+iLyHRHpKrT5PCIyRUQ2iMiO+O/h\nn5+vvdOJgjSmiPvYNuAjwCavA8lUgU/T/y6w2usgciACfE5VFwJXA58+3/8zpxNFmlPEfUlVW1V1\nl9dxZEnBTtNX1U1At9dxZJuqtqvq6/HHJxm9ItmSrL3TieIcnwDsfmduSjRNP+kvnXGLiEwHLgde\nSdbG8ztc5XuKeD6l89mM8ZKIVAGPAX+hqr3J2nmeKLIwRdxZqT5bAbFp+j4kIiWMJonvq+pPztfW\n6VOPNKeIG+/ZNH2fEREBvg20quo/pWrvdKIgyRTxQiAivysih4BrgCdE5BmvY7pYGU7Td5qIPAq8\nBMwTkUMico/XMWXJdcDHgBvix9YbIrImWWObwm2MScn1EYUxxgGWKIwxKVmiMMakZInCGJOSJQpj\nTEqWKIwxKVmiMMak9P8B5cS7YUOAvQYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fac3a03bb70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots()\n",
    "original_quad = Polygon(original_matrix, True)\n",
    "new_quad = Polygon(transformed_matrix, True)\n",
    "patch = PatchCollection([original_quad, new_quad], \n",
    "                        cmap=matplotlib.cm.jet,\n",
    "                        alpha=0.2)\n",
    "axes.add_collection(patch)\n",
    "_ = axes.plot(line_x, \n",
    "              grad_line_1, \n",
    "              dashes=[5, 5])\n",
    "_ = axes.plot(line_x, \n",
    "              grad_line_2, \n",
    "              dashes=[5, 5])\n",
    "_ = axes.set_xlim([-2, 2])\n",
    "_ = axes.set_ylim([-2, 2])\n",
    "plt.gca().set_aspect('equal', adjustable='box')"
   ]
  }
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