Module_3a_linear_algebra_eigenvectors.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Intermediate Linear Algebra  - Eigenvalues & Eigenvectors\n",
    "\n",
    "### Key Equation: $Ax = \\lambda b ~~ \\text{for} ~~ n \\times n $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transformations\n",
    "\n",
    "So what really happens when we multiply the $A$ matrix with a vector $x$\n",
    "\n",
    "Lets say we have a vector - $x$\n",
    "\n",
    "$$ x = \\begin{bmatrix} -1 \\\\ 1 \\end{bmatrix} $$\n",
    "\n",
    "What happens when we multiply by a matrix - $A$\n",
    "\n",
    "$$ A = \\begin{bmatrix} 6 & 2 \\\\ 2 & 6 \\end{bmatrix} $$\n",
    "\n",
    "$$ Ax = \\begin{bmatrix} 6 & 2 \\\\ 2 & 6 \\end{bmatrix} \\begin{bmatrix} -1 \\\\ 1 \\end{bmatrix} = \\begin{bmatrix} -4  \\\\ 4 \\end{bmatrix} $$\n",
    "\n",
    "$$ Ax = 4Ix $$\n",
    "\n",
    "$$ Ax = 4x $$\n",
    "\n",
    "So this particular matrix has just scaled our original vector. It is a scalar transformation. Other matrices can do reflection, rotation and any arbitary transformation in the same 2d space for n = 2.\n",
    "\n",
    "Lets see what has happened through code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "plt.style.use('fivethirtyeight')\n",
    "plt.rcParams['figure.figsize'] = (10, 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def vector_plot (vector):\n",
    "    X,Y,U,V = zip(*vector)\n",
    "    C = [1,1,2,2]\n",
    "    plt.figure()\n",
    "    ax = plt.gca()\n",
    "    ax.quiver(X,Y,U,V,C, angles='xy',scale_units='xy',scale=1)\n",
    "    ax.set_xlim([-6,6])\n",
    "    ax.set_ylim([-6,6])\n",
    "    plt.axhline(0, color='grey', linewidth=1)\n",
    "    plt.axvline(0, color='grey', linewidth=1)\n",
    "    plt.axes().set_aspect('equal')\n",
    "    plt.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "A = np.array([[ 6 , 2],\n",
    "               [ 2 , 6]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([[-1],\n",
    "             [1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "v = A.dot(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# All the vectors start at 0, 0 \n",
    "vAX = np.r_[[0,0],A[:,0]]\n",
    "vAY = np.r_[[0,0],A[:,1]]\n",
    "vx = np.r_[[0,0],x[:,0]]\n",
    "vv = np.r_[[0,0],v[:,0]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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1OpygL+Sv/7OtIya3jZcuRqFs9TZ9+f+ONPk6UXC/+H4GAwY1/nuVWLtK/vAh1eu7StBx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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111a9c828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vector_plot([vAX, vAY, vx, vv])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  Solving Equation $Ax=\\lambda x$\n",
    "\n",
    "\n",
    "\n",
    "### Special Case: $Ax = 0$ \n",
    "\n",
    "So far we have been solving the equation $Ax = b$. Let us just look at special case when $b=0$.\n",
    "\n",
    "$$ Ax =0 $$\n",
    "\n",
    "If $A^{-1}$ exists (the matrix is non-singular and invertable), then the solution is trival\n",
    "\n",
    "$$ A^{-1}Ax =0 $$\n",
    "$$ x = 0$$\n",
    "\n",
    "If $A^{-1}$ does not exist, then there may be infinitely many other solutions $x$. And since $A^{-1}$ is a singular matrix then\n",
    "\n",
    "$$||A|| = 0 $$\n",
    "\n",
    "### General Case\n",
    "\n",
    "\n",
    "The second part of linear algebra is solving the equation, for a given $A$ - \n",
    "\n",
    "$$ Ax = \\lambda x$$\n",
    "\n",
    "Note that both $x$ and $\\lambda$ are unknown in this equation. For all solutions of them:\n",
    "\n",
    "$$ \\text{eigenvalues} = \\lambda $$\n",
    "$$ \\text{eigenvectors} = x $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculating Eigenvalues\n",
    "\n",
    "So let us first solve this for $\\lambda$ :\n",
    "\n",
    "$$ Ax = \\lambda Ix $$\n",
    "\n",
    "$$ (A-\\lambda I)x = 0 $$\n",
    "\n",
    "So for non-trivial solution of $x$, $A$ should be singular:\n",
    "\n",
    "$$ ||A - \\lambda I|| = 0 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## For 2 x 2 Matrix\n",
    "\n",
    "Let us use the sample $A$ vector:\n",
    "\n",
    "$$ A =  \\begin{bmatrix}3 & 1\\\\ 1 & 3\\end{bmatrix} $$\n",
    "\n",
    "So our equation becomes: \n",
    "\n",
    "$$ \\begin{bmatrix}3 & 1\\\\ 1 & 3\\end{bmatrix} \\begin{bmatrix}x \\\\ y\\end{bmatrix} = \\begin{bmatrix}\\lambda & 0\\\\ 0 & \\lambda \\end{bmatrix} \\begin{bmatrix}x \\\\ y\\end{bmatrix} $$\n",
    "\n",
    "$$ \\begin{bmatrix}3 - \\lambda & 1\\\\ 1 & 3 - \\lambda \\end{bmatrix} \\begin{bmatrix}x \\\\ y\\end{bmatrix} = 0 $$\n",
    "\n",
    "So for a singular matrix: \n",
    "\n",
    "$$ \\begin{Vmatrix}3 - \\lambda & 1\\\\ 1 & 3 - \\lambda \\end{Vmatrix} = 0 $$\n",
    "\n",
    "\n",
    "$$ (3 - \\lambda)^2 - 1  = 0 $$\n",
    "\n",
    "$$ \\lambda^2 - 6\\lambda + 8 = 0 $$\n",
    "\n",
    "$$ (\\lambda - 4)(\\lambda - 2) = 0 $$\n",
    "\n",
    "$$ \\lambda_1 = 2, \\lambda_2 = 4 $$\n",
    "\n",
    "$$||A|| = \\lambda_{1} \\lambda_{2} $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Calculating Eigenvectors\n",
    "\n",
    "For $\\lambda = 2$,\n",
    "\n",
    "$$ \\begin{bmatrix}3 - \\lambda & 1\\\\ 1 & 3 - \\lambda \\end{bmatrix} \\begin{bmatrix}x \\\\ y\\end{bmatrix} = \\begin{bmatrix}1 & 1\\\\ 1 & 1 \\end{bmatrix} \\begin{bmatrix}x \\\\ y\\end{bmatrix} = 0 $$\n",
    "\n",
    "So one simple solution is:\n",
    "\n",
    "$$ \\begin{bmatrix}x \\\\ y\\end{bmatrix} = \\begin{bmatrix}-1 \\\\ 1\\end{bmatrix} $$\n",
    "\n",
    "For $\\lambda = 4$,\n",
    "\n",
    "$$ \\begin{bmatrix}3 - \\lambda & 1\\\\ 1 & 3 - \\lambda \\end{bmatrix} \\begin{bmatrix}x \\\\ y\\end{bmatrix} = \\begin{bmatrix}-1 & 1\\\\ 1 & -1 \\end{bmatrix} \\begin{bmatrix}x \\\\ y\\end{bmatrix} = 0 $$\n",
    "\n",
    "So one simple solution is:\n",
    "\n",
    "$$ \\begin{bmatrix}x \\\\ y\\end{bmatrix} = \\begin{bmatrix}1 \\\\ 1\\end{bmatrix} $$\n",
    "\n",
    "\n",
    "The eigenvectors are orthonormal to each other in this case.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vector Representation (2x2)\n",
    "\n",
    "A vector representation for this is now:\n",
    "\n",
    "$$ \\begin{bmatrix}3 \\\\ 1\\end{bmatrix} x + \\begin{bmatrix}1 \\\\ 3\\end{bmatrix} y = \\begin{bmatrix} \\lambda \\\\  0 \\end{bmatrix} x + \\begin{bmatrix} 0 \\\\ \\lambda \\end{bmatrix} y $$\n",
    "\n",
    "Now we need to draw these vectors and see the result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([[ 3 , 1],\n",
    "               [ 1 , 3]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "eigen_val, eigen_vec = np.linalg.eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 4.,  2.])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigen_val"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.70710678, -0.70710678],\n",
       "       [ 0.70710678,  0.70710678]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigen_vec"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.70710678,  0.70710678])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigen_vec[:,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# All the vectors start at 0, 0 \n",
    "vX1 = np.r_[[0,0],A[:,0]]\n",
    "vY1 = np.r_[[0,0],A[:,1]]\n",
    "vE1 = np.r_[[0,0],eigen_vec[:,0]] * 2\n",
    "vE2 = np.r_[[0,0],eigen_vec[:,1]] * 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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a9Z0OOnuysuExRAHRwPJA1NbW6uDBg5o+fXqj48OHD9e+ffusXg4IixsjEecNNhy/HodM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\nMPpfrN5HzHc2jAAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111a9c240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vector_plot([vX1, vY1, vE1, vE2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3 x 3 Matrix\n",
    "\n",
    "\n",
    "Let us write it in the form \n",
    "\n",
    "$$ Ax = \\lambda x $$\n",
    "\n",
    "$$ \\begin{bmatrix}1 & 1 & 1 \\\\ 3 & 8 & 1 \\\\ 5 & -4 & 3\\end{bmatrix}\\begin{bmatrix} x \\\\y \\\\ z\\end{bmatrix}= \\lambda \\begin{bmatrix} x\\\\ y \\\\ x \\end{bmatrix} $$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f = np.matrix([[1,1,1],\n",
    "               [3,8,1],\n",
    "               [5,-4,3]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-0.70156212,  5.70156212,  7.        ]),\n",
       " matrix([[ -5.44055877e-01,   8.18568585e-02,   1.73092434e-17],\n",
       "         [  9.17246072e-02,  -4.85526254e-01,  -7.07106781e-01],\n",
       "         [  8.34020263e-01,   8.70381360e-01,   7.07106781e-01]]))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eig(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises on Eigenvalues and Eigenvectors\n",
    "\n",
    "$$ U =  \\begin{bmatrix}1 & 1 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & -1 & 4\\end{bmatrix}$$\n",
    "\n",
    "$$ S =  \\begin{bmatrix}3 & 1 & 2 \\\\ 1 & 4 & 5 \\\\ 2 & 5 & 6 \\end{bmatrix}$$\n",
    "\n",
    "$$ T =  \\begin{bmatrix}5 & 8 \\\\ 0 & 5 \\end{bmatrix}$$\n",
    "\n",
    "\n",
    "Write the matrix as np.matrix and find the Eigenvalues and Eigenvectors?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
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    "## Symmetric Matrix and Eigen Vectors\n",
    "\n",
    "$$ V =  \\begin{bmatrix}1 & 2 & -1 \\\\ 2 & -1 & -2 \\\\ -1 & -2 & 1\\end{bmatrix}$$\n"
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