{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter3 Orthogonality"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.1.1 Pg: 143"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x1=\n",
"[[ 2]\n",
" [ 2]\n",
" [-1]]\n",
"x2=\n",
"[[-1]\n",
" [ 2]\n",
" [ 2]]\n",
"x1'*x2=\n",
"[[0]]\n",
"Therefore,X1 is orthogonal to x2 .Both have length of sqrt(14).\n"
]
}
],
"source": [
"from numpy import mat,transpose\n",
"x1=mat([[2],[2],[-1]])\n",
"print 'x1=\\n',x1\n",
"x2=mat([[-1],[2],[2]])\n",
"print 'x2=\\n',x2\n",
"print \"x1'*x2=\\n\",(transpose(x1)*x2)\n",
"print 'Therefore,X1 is orthogonal to x2 .Both have length of sqrt(14).'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.1.3 Pg: 145"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A=\n",
"[[1 3]\n",
" [2 6]\n",
" [3 9]]\n",
"Null space=\n",
"[[-0.9486833 ]\n",
" [ 0.31622777]]\n",
"A(1,:)*ns= [[ -2.22044605e-16]]\n",
"A(2,:)*ns= [[ -4.44089210e-16]]\n",
"A(3,:)*ns= [[ -4.44089210e-16]]\n",
"This shows that the null space of A is orthogonal to the row space.\n"
]
}
],
"source": [
"from numpy import mat,linalg,atleast_2d\n",
"A=mat([[1, 3],[2, 6],[3, 9]])\n",
"print 'A=\\n',A\n",
"def nullspace(A, atol=1e-13, rtol=0):\n",
" \n",
" A = atleast_2d(A)\n",
" u, s, vh = linalg.svd(A)\n",
" tol = max(atol, rtol * s[0])\n",
" nnz = (s >= tol).sum()\n",
" ns = vh[nnz:].conj().T\n",
" return ns\n",
"ns=nullspace(A)\n",
"print 'Null space=\\n',ns\n",
"print 'A(1,:)*ns=',(A[0]*ns)\n",
"print 'A(2,:)*ns=',(A[1]*ns)\n",
"print 'A(3,:)*ns=',(A[2]*ns)\n",
"print 'This shows that the null space of A is orthogonal to the row space.'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"## Ex:3.2.1 Pg: 155"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"b=\n",
"[[1]\n",
" [2]\n",
" [3]]\n",
"a=\n",
"[[1]\n",
" [1]\n",
" [1]]\n",
"Projection p of b onto the line through a is x***a=\n",
"[[2]\n",
" [2]\n",
" [2]]\n",
"cos(thetha) = 0.925820099773\n"
]
}
],
"source": [
"from numpy import mat,transpose,sqrt,transpose\n",
"b=mat([[1],[2],[3]])\n",
"print 'b=\\n',b\n",
"a=mat([[1],[1],[1]])\n",
"print 'a=\\n',a\n",
"x=(transpose(a)*b)/(transpose(a)*a)\n",
"x=x[0,0]\n",
"print 'Projection p of b onto the line through a is x***a=\\n',(x*a)\n",
"cos_theta=(transpose(a)*b)[0,0]/(sqrt(transpose(a)*a)[0,0]*sqrt(transpose(b)*b)[0,0])\n",
"print 'cos(thetha) =',cos_theta"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.2.2 Pg: 156"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"a=\n",
"[[ 1.]\n",
" [ 1.]\n",
" [ 1.]]\n",
"Matrix that projects onto a line through a=(1,1,1) is\n",
"[[ 0.33333333 0.33333333 0.33333333]\n",
" [ 0.33333333 0.33333333 0.33333333]\n",
" [ 0.33333333 0.33333333 0.33333333]]\n"
]
}
],
"source": [
"from numpy import mat,transpose\n",
"a=mat([[1.],[1],[1]])\n",
"print 'a=\\n',a\n",
"P=(a*transpose(a))/(transpose(a)*a)[0,0]\n",
"print 'Matrix that projects onto a line through a=(1,1,1) is\\n',P"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.2.3 Pg: 156"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"a=\n",
"[[ 0.70710678]\n",
" [ 0.70710678]]\n",
"Projection of line onto the thetha-direction(thetha taken as 45) in the x-y plane passing through a is\n",
"[[ 0.5 0.5]\n",
" [ 0.5 0.5]]\n"
]
}
],
"source": [
"from numpy import mat,transpose,sin,pi,cos\n",
"thetha=45# #Taking some value for thetha\n",
"a=mat([[cos(thetha*pi/180)],[sin(thetha*pi/180)]])\n",
"print 'a=\\n',a\n",
"P=(a*transpose(a))/(transpose(a)*a)\n",
"print 'Projection of line onto the thetha-direction(thetha taken as 45) in the x-y plane passing through a is\\n',P"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.3.1 Pg: 165"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A=\n",
"[[ 0.09246118 0.04909557 0.71167391 0.77035248]\n",
" [ 0.4079182 0.15730679 0.91259156 0.21064709]\n",
" [ 0.02898159 0.87211114 0.11852139 0.69009963]\n",
" [ 0.16889615 0.35412933 0.60067694 0.1180308 ]]\n",
"P=A*inv(A*A)*A\n",
"Projection of a invertible 4x4 matrix on to the whole space is:\n",
"[[ 3.38771356e-02 -8.74775251e-02 -1.51348916e-02 1.06873528e+00]\n",
" [ -8.74775251e-02 1.39664206e-03 1.08138671e+00 4.69417392e-03]\n",
" [ -1.51348916e-02 1.08138671e+00 -1.24018336e-04 -6.61277992e-02]\n",
" [ 1.06873528e+00 4.69417392e-03 -6.61277992e-02 -7.30165590e-03]]\n",
"Its identity matrix.\n"
]
}
],
"source": [
"from numpy import mat,transpose,linalg,random\n",
"A=random.rand(4,4)\n",
"print 'A=\\n',A\n",
"P=A*linalg.inv(transpose(A)*A)*transpose(A)\n",
"print 'P=A*inv(A''*A)*A'\n",
"print 'Projection of a invertible 4x4 matrix on to the whole space is:\\n',P\n",
"print 'Its identity matrix.'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.3.2 Pg: 166"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"b=C+Dt\n",
"Ax=b\n",
"A=\n",
"[[ 1 -1]\n",
" [ 1 1]\n",
" [ 1 2]]\n",
"b=\n",
"[[1]\n",
" [1]\n",
" [3]]\n",
"If Ax=b could be solved then they would be no errors, they cant be solved because the points are not on a line.Therefore they are solved by least squares.\n",
"so,AAx**=Ab\n",
"C** = 1.28571428571\n",
"D**= 0.571428571429\n",
"The best line is 9/7+4/7t\n"
]
}
],
"source": [
"from numpy import mat,transpose,zeros,linalg\n",
"print 'b=C+Dt'\n",
"print 'Ax=b'\n",
"A=mat([[1, -1],[1, 1],[1, 2]])\n",
"print 'A=\\n',A\n",
"b=mat([[1],[1],[3]])\n",
"print 'b=\\n',b\n",
"print 'If Ax=b could be solved then they would be no errors, they can''t be solved because the points are not on a line.Therefore they are solved by least squares.'\n",
"print 'so,A''Ax**=A''b'\n",
"x=zeros([1,2])\n",
"x=linalg.solve((transpose(A)*A), (transpose(A)*b))\n",
"print 'C** =',x[0,0]\n",
"print 'D**=',x[1,0]\n",
"print 'The best line is 9/7+4/7t'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.4.1 Pg: 175"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Q=\n",
"[[ 0.70710678 -0.70710678]\n",
" [ 0.70710678 0.70710678]]\n",
"\n",
"Q'=inv(Q)=\n",
"[[ 0.70710678 0.70710678]\n",
" [-0.70710678 0.70710678]]\n",
"\n",
"Q rotates every vector through an angle thetha, and Q rotates it back through -thetha.The columns are clearly orthogonal and they are orthonormal because sin**2(theta)+cos**2(thetha)=1.\n"
]
}
],
"source": [
"from numpy import mat,transpose,sin,pi,cos\n",
"thetha=45##Taking some value for thetha.\n",
"Q=mat([[cos(pi/180*thetha),-sin(thetha*pi/180)],[sin(pi/180*thetha),cos(pi/180*thetha)]])\n",
"print 'Q=\\n',Q\n",
"print \"\\nQ'=inv(Q)=\\n\",transpose(Q)\n",
"print '\\nQ rotates every vector through an angle thetha, and Q'' rotates it back through -thetha.The columns are clearly orthogonal and they are orthonormal because sin**2(theta)+cos**2(thetha)=1.'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.4.2 Pg: 175"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Any permutation matrix is an orthogonal matrix.The columns are certainly unit vectors and certainly orthogonal-because the 1 appears in a differnt place in each column\n",
"P=\n",
"[[0 1 0]\n",
" [0 0 1]\n",
" [1 0 0]]\n",
"inv(P)=P'=\n",
"[[0 0 1]\n",
" [1 0 0]\n",
" [0 1 0]]\n",
"And,P'*P=\n",
"[[1 0 0]\n",
" [0 1 0]\n",
" [0 0 1]]\n"
]
}
],
"source": [
"from numpy import mat,transpose\n",
"print 'Any permutation matrix is an orthogonal matrix.The columns are certainly unit vectors and certainly orthogonal-because the 1 appears in a differnt place in each column'\n",
"P=mat([[0, 1, 0],[0, 0 ,1],[1, 0, 0]])\n",
"print 'P=\\n',P\n",
"print \"inv(P)=P'=\\n\",transpose(P)\n",
"print \"And,P'*P=\\n\",(transpose(P)*P)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.4.3 Pg: 175"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"If we project b=(x,y,z) onto the x-y plane then its projection is p=(x,y,0),and is the sum of projection onto x- any y-axes.\n",
"q1=\n",
"[[1]\n",
" [0]\n",
" [0]]\n",
"q2=\n",
"[[0]\n",
" [1]\n",
" [0]]\n",
"Overall projection matrix,P=\n",
"[[1 0 0]\n",
" [0 1 0]\n",
" [0 0 0]]\n",
"and,P[x#y#z]=[x#y#0]\n",
"Projection onto a plane=sum of projections onto orthonormal q1 and q2.\n"
]
}
],
"source": [
"from numpy import mat,transpose,random\n",
"print 'If we project b=(x,y,z) onto the x-y plane then its projection is p=(x,y,0),and is the sum of projection onto x- any y-axes.'\n",
"b=random.rand(3,1)\n",
"q1=mat([[1],[0],[0]])\n",
"print 'q1=\\n',q1\n",
"q2=mat([[0],[1],[0]])\n",
"print 'q2=\\n',q2\n",
"P=q1*transpose(q1)+q2*transpose(q2)\n",
"print 'Overall projection matrix,P=\\n',P\n",
"print 'and,P[x#y#z]=[x#y#0]'\n",
"print 'Projection onto a plane=sum of projections onto orthonormal q1 and q2.'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.4.4 Pg: 166"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"y=C+Dt\n",
"Ax=b\n",
"A=\n",
"[[ 1 -3]\n",
" [ 1 0]\n",
" [ 1 3]]\n",
"y=\n",
"[[ 0.72089857]\n",
" [ 0.89298883]\n",
" [ 0.60457288]]\n",
"the columns of A are orthogonal,so\n",
"C** =\n",
"[[ 0.12324779]]\n",
"D** =\n",
"[[-0.12014976 0. 0.12014976]\n",
" [-0.14883147 0. 0.14883147]\n",
" [-0.10076215 0. 0.10076215]]\n",
"C** gives the besy fit ny horizontal line, whereas D**t is the best fit by a straight line through the origin.\n"
]
}
],
"source": [
"from numpy import mat,transpose,random,zeros\n",
"print 'y=C+Dt'\n",
"print 'Ax=b'\n",
"A=mat([[1, -3],[1, 0],[1, 3]])\n",
"print 'A=\\n',A\n",
"y=random.rand(3,1)\n",
"print 'y=\\n',y\n",
"print 'the columns of A are orthogonal,so'\n",
"x=zeros([1,2])\n",
"print \"C** =\\n\",( (mat([1, 1, 1])*y)/(transpose(A[:,1])*A[:,1]) )\n",
"\n",
"print \"D** =\\n\",( mat([-3, 0 ,3]*y)/(transpose(A[:,1])*A[:,1]) )\n",
"print 'C** gives the besy fit ny horizontal line, whereas D**t is the best fit by a straight line through the origin.'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:3.4.5 Pg: 166"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A=\n",
"[[1 0 1]\n",
" [1 0 0]\n",
" [2 1 0]]\n",
"Q=\n",
"[[ 0.40824829 0. 0.91287093]\n",
" [ 0.40824829 0. -0.18257419]\n",
" [ 0.81649658 1. -0.36514837]]\n"
]
}
],
"source": [
"from numpy import mat,transpose,zeros,shape,linalg\n",
"A=mat([[1, 0 ,1],[1, 0, 0],[2, 1, 0]]) #independent vectors stored in columns of A\n",
"print 'A=\\n',A\n",
"m,n=shape(A)\n",
"V=mat(zeros([n,n]))\n",
"R=mat(zeros([n,n]))\n",
"for k in range(0,n):\n",
" V[:,k]=A[:,k]\n",
" for j in range(0,k-1):\n",
" R[j,k]=transpose(V[:,j])*A[:,k]\n",
" V[:,k]=V[:,k]-R[j,k]*V[:,j]\n",
" \n",
" R[k,k]=linalg.norm(V[:,k])\n",
" V[:,k]=V[:,k]/R[k,k]\n",
"\n",
"print 'Q=\\n',V"
]
}
],
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