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+{
+ "cells": [
+ {
+ "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)))"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 2",
+ "language": "python",
+ "name": "python2"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 2
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython2",
+ "version": "2.7.9"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
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