{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 5 Eigenvalues and Eigenvectors"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:5.1.1 Pg: 238"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Eigen values: [ 3. 2.]\n",
"Eigen vectors:\n",
"[[ 1. 0.]] \n",
" and\n",
"[[ 0. 1.]]\n"
]
}
],
"source": [
"from numpy import mat\n",
"from numpy.linalg import eig\n",
"A=mat([[3, 0],[0, 2]])\n",
"Eig,V=eig(A)\n",
"print 'Eigen values:',Eig\n",
"print 'Eigen vectors:\\n',V[0],'\\n and\\n',V[1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:5.1.2 Pg: 238"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The eigen values of a projection matrix are 1 or 0.\n",
"Eigen values: ['1', '0']\n",
"Eigen vectors:\n",
"[[ 0.70710678 -0.70710678]] \n",
" and\n",
"[[ 0.70710678 0.70710678]]\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import mat\n",
"from numpy.linalg import eig\n",
"print 'The eigen values of a projection matrix are 1 or 0.'\n",
"P=mat([[1/2, 1/2],[1/2, 1/2]])\n",
"Eig,V=eig(P)\n",
"Eig=[\"%.f\"%xx for xx in Eig]\n",
"print 'Eigen values:',Eig\n",
"print 'Eigen vectors:\\n',V[0],'\\n and\\n',V[1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:5.2.1 Pg: 238"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Eigenvalue matrix: [ 3. 2.]\n",
"S=\n",
"[[ 1. 0.]\n",
" [ 0. 1.]]\n",
"AS=S*eigenvaluematrix\n",
"[[ 3. 0.]\n",
" [ 0. 2.]]\n",
"Therefore inv(S)*A*S=eigenvalue matrix\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import mat\n",
"from numpy.linalg import eig\n",
"P=mat([[1/2, 1/2],[1/2, 1/2]])\n",
"Val,V=eig(A)\n",
"print 'Eigenvalue matrix:',Val\n",
"print 'S=\\n',V\n",
"print 'AS=S*eigenvaluematrix\\n',(A*V)\n",
"print 'Therefore inv(S)*A*S=eigenvalue matrix'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:5.2.2 Pg: 238"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The eigenvalues themselves are not so clear for a rotation.\n",
"90 degree rotation\n",
"K= [[ 0 -1]\n",
" [ 1 0]]\n",
"Eigen values: [ 0.+1.j 0.-1.j]\n",
"Eigen vectors:\n",
"[[ 0.70710678+0.j 0.70710678-0.j]] \n",
" and\n",
"[[ 0.-0.70710678j 0.+0.70710678j]]\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import mat\n",
"from numpy.linalg import eig\n",
"print 'The eigenvalues themselves are not so clear for a rotation.'\n",
"print '90 degree rotation'\n",
"K=mat([[0, -1],[1, 0]])\n",
"print 'K=',K\n",
"Val,V=eig(K)\n",
"print 'Eigen values:',Val\n",
"print 'Eigen vectors:\\n',V[0],'\\n and\\n',V[1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:5.2.3 Pg: 249"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"K is rotation through 90 degree,then K**2 is rotation through 180 degree and inv(k is rotation through -90 degree)\n",
"K=\n",
"[[ 0 -1]\n",
" [ 1 0]]\n",
"K**2=\n",
"[[-1 0]\n",
" [ 0 -1]]\n",
"K**3=\n",
"[[ 0 1]\n",
" [-1 0]]\n",
"K**4=\n",
"[[1 0]\n",
" [0 1]]\n",
"K**4 is a complete rotation through 360 degree.\n",
"Eigen value matrix,D of K:\n",
"[ 0.+1.j 0.-1.j]\n",
"and also D**4=\n",
"[ 1.+0.j 1.+0.j]\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import mat\n",
"from numpy.linalg import eig\n",
"print 'K is rotation through 90 degree,then K**2 is rotation through 180 degree and inv(k is rotation through -90 degree)'\n",
"K=mat([[0, -1],[1, 0]])\n",
"print 'K=\\n',K\n",
"print 'K**2=\\n',(K*K)\n",
"print 'K**3=\\n',(K*K*K)\n",
"print 'K**4=\\n',(K**4)\n",
"D,V=eig(K)\n",
"print 'K**4 is a complete rotation through 360 degree.'\n",
"print 'Eigen value matrix,D of K:\\n',D\n",
"print 'and also D**4=\\n',(D**4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:5.3.1 Pg: 249"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A=\n",
"[[ 0. 4. ]\n",
" [ 0. 0.5]]\n",
"Eigen values: [ 0. 0.5]\n",
"[[ 1.]\n",
" [ 0.]]\n",
"k= 0\n",
"U(k+1)(K from 0 to 5)\n",
"[[ 0.]\n",
" [ 0.]]\n",
"k= 1\n",
"U(k+1)(K from 0 to 5)\n",
"[[ 0.]\n",
" [ 0.]]\n",
"k= 2\n",
"U(k+1)(K from 0 to 5)\n",
"[[ 0.]\n",
" [ 0.]]\n",
"k= 3\n",
"U(k+1)(K from 0 to 5)\n",
"[[ 0.]\n",
" [ 0.]]\n",
"k= 4\n",
"U(k+1)(K from 0 to 5)\n",
"[[ 0.]\n",
" [ 0.]]\n",
"k= 5\n",
"U(k+1)(K from 0 to 5)\n",
"[[ 0.]\n",
" [ 0.]]\n",
"k= 5\n",
"U(k+1)=\n",
"[[ 0.49613894]\n",
" [ 0.06201737]]\n",
"k= 5\n",
"U(k+1)=\n",
"[[ 0.24806947]\n",
" [ 0.03100868]]\n",
"k= 5\n",
"U(k+1)=\n",
"[[ 0.12403473]\n",
" [ 0.01550434]]\n",
"k= 5\n",
"U(k+1)=\n",
"[[ 0.06201737]\n",
" [ 0.00775217]]\n",
"k= 5\n",
"U(k+1)=\n",
"[[ 0.03100868]\n",
" [ 0.00387609]]\n",
"k= 5\n",
"U(k+1)=\n",
"[[ 0.01550434]\n",
" [ 0.00193804]]\n"
]
}
],
"source": [
"from __future__ import division\n",
"from numpy import mat\n",
"from numpy.linalg import eig\n",
"A=mat([[0, 4],[0, 1/2]])\n",
"print 'A=\\n',A\n",
"Eig,zz=eig(A)\n",
"print 'Eigen values:',Eig\n",
"D,v=eig(A)\n",
"u0=v[:,0] #Taking u0 as the 1st eigen Vector.\n",
"print u0\n",
"for k in range(0,6):\n",
" print 'k=',k\n",
" u=A*u0\n",
" print 'U(k+1)(K from 0 to 5)\\n',u\n",
" u0=u\n",
"\n",
"u0=v[:,1] ##Taking u0 as the 2nd eigen vector.\n",
"for ki in range(0,6):\n",
" print 'k=',k\n",
" u=A*u0\n",
" print 'U(k+1)=\\n',u\n",
" u0=u"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:5.5.1 Pg:282"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x= (3+4j)\n",
"xx_= (25+0j)\n",
"r= (5+0j)\n"
]
}
],
"source": [
"from numpy import sqrt\n",
"x=3+4*1J\n",
"print 'x=',x\n",
"x_=x.conjugate()\n",
"print 'xx_=',x*x_\n",
"r=sqrt(x*x_)\n",
"print 'r=',r"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:5.5.2 Pg:282"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Length of x squared: 2.0\n",
"Length of y squared: 25.0\n"
]
}
],
"source": [
"from numpy import mat\n",
"i=1J\n",
"x=mat([[1, i]]).H\n",
"y=mat([[2+1*i, 2-4*i]]).H\n",
"print 'Length of x squared:',abs(x.H*x)[0,0]\n",
"print 'Length of y squared:',abs( y.H*y)[0,0]"
]
}
],
"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
}