{
 "cells": [
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 6 Positive Definite Matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex:6.1.1 Pg:313"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(x,y)=x**2-10*x*y+y**2\n",
      "f(1,1)= -8\n",
      "The conditions a>0 and c>0 ensure that f(x,y) is positive on the x and y axes. But this function is negative on the line x=y,because b=-10 overwhelms a and c. \n"
     ]
    }
   ],
   "source": [
    "print 'f(x,y)=x**2-10*x*y+y**2'\n",
    "a=1\n",
    "c=1\n",
    "def f(x,y):\n",
    "    ff=x**2-10*x*y+y**2\n",
    "    return ff\n",
    "print 'f(1,1)=',f(1,1)\n",
    "print 'The conditions a>0 and c>0 ensure that f(x,y) is positive on the x and y axes. But this function is negative on the line x=y,because b=-10 overwhelms a and c. '"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex:6.1.3 Pg:315"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(x,y)=2*x**2+4*x*y+y**2\n",
      "ac= 1\n",
      "b**2= 4\n",
      "Saddle point,as ac<b**2\n"
     ]
    }
   ],
   "source": [
    "print 'f(x,y)=2*x**2+4*x*y+y**2'\n",
    "A=[[2, 2],[2, 1]]\n",
    "a=1\n",
    "c=1\n",
    "b=2\n",
    "print 'ac=',a*c\n",
    "print 'b**2=',b**2\n",
    "print 'Saddle point,as ac<b**2'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex:6.1.4 Pg:315"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(x,y)=2*x**2+4*x*y+y**2\n",
      "ac= 0\n",
      "b**2= 1\n",
      "Saddle point,as ac<b**2\n"
     ]
    }
   ],
   "source": [
    "print 'f(x,y)=2*x**2+4*x*y+y**2'\n",
    "A=[[2 ,2],[2, 1]]\n",
    "a=0\n",
    "c=0\n",
    "b=1\n",
    "print 'ac=',a*c\n",
    "print 'b**2=',b**2\n",
    "print 'Saddle point,as ac<b**2'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex:6.2.2 Pg:313"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(x,y)=x**2+4*x*y+y**2\n",
      "f(0,0)= 0\n",
      "Here 2b=4  it still does not ensure a minimum ,the sign of b is of no importance.Neither F nor f has a minimum at(0,0) because f(1,-1)=-1.\n"
     ]
    }
   ],
   "source": [
    "print 'f(x,y)=x**2+4*x*y+y**2'\n",
    "a=1\n",
    "c=1\n",
    "def f(x,y):\n",
    "    ff=x**2+4*x*y+y**2\n",
    "    return ff\n",
    "print 'f(0,0)=',f(0,0)\n",
    "print 'Here 2b=4  it still does not ensure a minimum ,the sign of b is of no importance.Neither F nor f has a minimum at(0,0) because f(1,-1)=-1.'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex:6.3.1 Pg:332"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A= [[-1]\n",
      " [ 2]\n",
      " [ 2]]\n",
      "U= [[-0.33333333  0.66666667  0.66666667]\n",
      " [ 0.66666667  0.66666667 -0.33333333]\n",
      " [ 0.66666667 -0.33333333  0.66666667]]\n",
      "diagnol= [ 3.]\n",
      "V'= [[ 1.]]\n",
      "A=U*diagnol*V'=\n",
      "[[-1.  2.  2.]\n",
      " [ 2.  2. -1.]\n",
      " [ 2. -1.  2.]]\n"
     ]
    }
   ],
   "source": [
    "from scipy.linalg import svd\n",
    "from numpy import array,transpose\n",
    "A=transpose(array([[-1, 2, 2]]))\n",
    "print 'A=',A\n",
    "ans=svd(A)\n",
    "U=ans[0]\n",
    "diagnol=ans[1]\n",
    "V=ans[2]        \n",
    "print 'U=',U\n",
    "print 'diagnol=',diagnol\n",
    "print \"V'=\",V\n",
    "print \"A=U*diagnol*V'=\\n\",U*diagnol*transpose(V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex:6.3.2 Pg:332"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A=\n",
      "[[-1  1  0]\n",
      " [ 0 -1  1]]\n",
      "U=\n",
      "[[-0.70710678  0.70710678]\n",
      " [ 0.70710678  0.70710678]]\n",
      "Diagonal=\n",
      "[[ 1.73205081  0.          0.        ]\n",
      " [ 0.          1.          0.        ]]\n",
      "V'= [[  4.08248290e-01  -7.07106781e-01   5.77350269e-01]\n",
      " [ -8.16496581e-01  -2.77555756e-16   5.77350269e-01]\n",
      " [  4.08248290e-01   7.07106781e-01   5.77350269e-01]]\n",
      "[[-1.22474487  0.70710678  0.        ]\n",
      " [ 1.22474487  0.70710678  0.        ]]\n",
      "A=U*diagonal*V'=\n",
      "[[-1.07735027  0.8660254  -0.29885849]\n",
      " [-0.07735027 -0.8660254   1.11535507]]\n"
     ]
    }
   ],
   "source": [
    "from scipy.linalg import svd\n",
    "from numpy import mat,array,transpose as tp, zeros\n",
    "A=mat([[-1, 1, 0],[0, -1, 1]])\n",
    "print 'A=\\n',A\n",
    "U,diagnl1,V=svd(A)\n",
    "diagnl=zeros([2,3])\n",
    "diagnl[0,0]=diagnl1[0]\n",
    "diagnl[1,1]=diagnl1[1]\n",
    "U=mat(U)\n",
    "diagnl=mat(diagnl)\n",
    "V=mat(V)\n",
    "print 'U=\\n',U\n",
    "print 'Diagonal=\\n',diagnl\n",
    "print \"V'=\",tp(V)\n",
    "print (U*diagnl)\n",
    "print \"A=U*diagonal*V'=\\n\",((U*diagnl)*tp(V))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex:6.3.3 Pg:332"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A=\n",
      "[[ 1 -2]\n",
      " [ 3 -1]]\n",
      "Q=\n",
      "[[  1.11022302e-16  -1.00000000e+00]\n",
      " [  1.00000000e+00   1.11022302e-16]]\n",
      "S=\n",
      "[[ 3. -1.]\n",
      " [-1.  2.]]\n",
      "A=SQ=\n",
      "[[ 1. -2.]\n",
      " [ 3. -1.]]\n"
     ]
    }
   ],
   "source": [
    "from numpy import mat, transpose as tp \n",
    "from scipy.linalg import svd\n",
    "A=mat([[1, -2],[3, -1]])\n",
    "print 'A=\\n',A\n",
    "U,S,V=svd(A)\n",
    "Q=U*tp(mat(V))\n",
    "S=V*S*tp(mat(V))\n",
    "print 'Q=\\n',Q\n",
    "print 'S=\\n',S\n",
    "print 'A=SQ=\\n',(Q*S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ex:6.3.4 Pg:332"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A=\n",
      "[[ 1 -2]\n",
      " [ 3 -1]]\n",
      "Q=\n",
      "[[  1.11022302e-16  -1.00000000e+00]\n",
      " [  1.00000000e+00   1.11022302e-16]]\n",
      "S=\n",
      "[[2 1]\n",
      " [1 3]]\n",
      "A=S'Q=\n",
      "[[-1.  2.]\n",
      " [-3.  1.]]\n"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "from numpy import mat, transpose as tp \n",
    "from scipy.linalg import svd\n",
    "\n",
    "A=mat([[1, -2],[3, -1]])\n",
    "print 'A=\\n',A\n",
    "U,diag1,V=svd(A)\n",
    "Q=U*tp(mat(V))\n",
    "S=mat([[2, 1],[1, 3]])\n",
    "print 'Q=\\n',Q\n",
    "print 'S=\\n',S\n",
    "print \"A=S'Q=\\n\",(S*tp(mat(Q)))"
   ]
  }
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