{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 1 - Matrix notation & matrix multiplication"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:1 Pg:20"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x=\n",
"[[u]\n",
" [v]\n",
" [w]]\n",
"R2=R2-R1,R3=R3-4*R1\n",
"[[1 1 1]\n",
" [0 0 3]\n",
" [0 2 4]]\n",
"R2<->R3\n",
"[[1 1 1]\n",
" [0 2 4]\n",
" [0 0 3]]\n",
"The system is now triangular and the equations can be solved by Back substitution\n"
]
}
],
"source": [
"from sympy.abc import u,v,w\n",
"import numpy as np\n",
"a =np.array([[1 ,1 ,1],[2, 2, 5],[4, 6, 8]])\n",
"x=[[u],[v],[w]]\n",
"x=np.array(x)\n",
"print 'x=\\n',x\n",
"print 'R2=R2-R1,R3=R3-4*R1'\n",
"a[1,:]=a[1,:]-2*a[0,:]\n",
"a[2,:]=a[2,:]-4*a[0,:]\n",
"print a\n",
"print 'R2<->R3'\n",
"import numpy as np\n",
"def swap_rows(arr, frm, to):\n",
" arr[[frm, to],:] = arr[[to, frm],:]\n",
"swap_rows(a,1,2)\n",
"print a\n",
"print 'The system is now triangular and the equations can be solved by Back substitution'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:2 Pg:21"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"a=\n",
"[[1 1 1]\n",
" [2 2 5]\n",
" [4 4 8]]\n",
"x=\n",
"[[u]\n",
" [v]\n",
" [w]]\n",
"R2=R2-2*R1,R3=R3-4*R1\n",
"[[1 1 1]\n",
" [0 0 3]\n",
" [0 0 4]]\n",
"No exchange of equations can avoid zero in the second pivot positon ,therefore the equations are unsolvable\n"
]
}
],
"source": [
"from sympy.abc import u,v,w\n",
"import numpy as np\n",
"a =np.array([[1, 1, 1],[2, 2, 5],[4, 4, 8]])\n",
"print 'a=\\n',a\n",
"x=[[u],[v],[w]]\n",
"x=np.array(x)\n",
"print 'x=\\n',x\n",
"print 'R2=R2-2*R1,R3=R3-4*R1'\n",
"a[1,:]=a[1,:]-2*a[0,:]\n",
"a[2,:]=a[2,:]-4*a[0,:]\n",
"print a\n",
"print 'No exchange of equations can avoid zero in the second pivot positon ,therefore the equations are unsolvable'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Ex:3 Pg:21"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \n",
"[[2 3]\n",
" [4 0]]\n",
"B = \n",
"[[ 1 2 0]\n",
" [ 5 -1 0]]\n",
"AB=\n",
"[[17 1 0]\n",
" [ 4 8 0]]\n"
]
}
],
"source": [
"import numpy as np\n",
"A=np.array([[2, 3],[4, 0]])\n",
"print 'A = \\n',A\n",
"B=np.array([[1, 2, 0],[5, -1, 0]])\n",
"print 'B = \\n',B\n",
"print 'AB=\\n',A.dot(B)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:4 Pg:22"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A=\n",
"[[2 3]\n",
" [7 8]]\n",
"P(Row exchange matrix)=\n",
"[[0 1]\n",
" [1 0]]\n",
"PA=\n",
"[[7 8]\n",
" [2 3]]\n"
]
}
],
"source": [
"import numpy as np\n",
"A=np.array([[2, 3],[7, 8]])\n",
"print 'A=\\n',A\n",
"P=np.array([[0, 1],[1, 0]])\n",
"print 'P(Row exchange matrix)=\\n',P\n",
"print 'PA=\\n',P.dot(A)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:5 Pg:24"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A=\n",
"[[1 2]\n",
" [3 4]]\n",
"I=\n",
"[[ 1. 0.]\n",
" [ 0. 1.]]\n",
"IA=\n",
"[[ 1. 2.]\n",
" [ 3. 4.]]\n"
]
}
],
"source": [
"import numpy as np\n",
"A=np.array([[1, 2],[3, 4]])\n",
"print 'A=\\n',A\n",
"I=np.identity(2)\n",
"print 'I=\\n',I\n",
"print 'IA=\\n',I.dot(A)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:6 Pg:25"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"E=\n",
"[[ 1. 0. 0.]\n",
" [-2. 1. 0.]\n",
" [ 0. 0. 1.]]\n",
"F=\n",
"[[ 1. 0. 0.]\n",
" [ 0. 1. 0.]\n",
" [ 1. 0. 1.]]\n",
"EF=\n",
"[[ 1. 0. 0.]\n",
" [-2. 1. 0.]\n",
" [ 1. 0. 1.]]\n",
"FE=\n",
"[[ 1. 0. 0.]\n",
" [-2. 1. 0.]\n",
" [ 1. 0. 1.]]\n",
"Here,EF=FE,so this shows that these two matrices commute\n"
]
}
],
"source": [
"import numpy as np\n",
"E=np.identity(3)\n",
"E[1,:]=E[1,:]-2*E[0,:]\n",
"print 'E=\\n',E\n",
"F=np.identity(3)\n",
"F[2,:]=F[2,:]+F[0,:]\n",
"print 'F=\\n',F\n",
"print 'EF=\\n',E.dot(F)\n",
"print 'FE=\\n',F.dot(E)\n",
"print 'Here,EF=FE,so this shows that these two matrices commute'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ex:7 Pg:25"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"E=\n",
"[[ 1. 0. 0.]\n",
" [-2. 1. 0.]\n",
" [ 0. 0. 1.]]\n",
"F=\n",
"[[ 1. 0. 0.]\n",
" [ 0. 1. 0.]\n",
" [ 1. 0. 1.]]\n",
"G=\n",
"[[ 1. 0. 0.]\n",
" [ 0. 1. 0.]\n",
" [ 0. 1. 1.]]\n",
"GE= [[ 1. 0. 0.]\n",
" [-2. 1. 0.]\n",
" [-2. 1. 1.]]\n",
"EG=\n",
"[[ 1. 0. 0.]\n",
" [-2. 1. 0.]\n",
" [ 0. 1. 1.]]\n",
"Here EG is not equal to GE,Therefore these two matrices do not commute and shows that most matrices do not commute.\n",
"GFE=\n",
"[[ 1. 0. 0.]\n",
" [-2. 1. 0.]\n",
" [-1. 1. 1.]]\n",
"EFG=\n",
"[[ 1. 0. 0.]\n",
" [-2. 1. 0.]\n",
" [ 1. 1. 1.]]\n",
"The product GFE is the true order of elimation.It is the matrix that takes the original A to the upper triangular U.\n"
]
}
],
"source": [
"import numpy as np\n",
"E=np.identity(3)\n",
"E[1,:]=E[1,:]-2*E[0,:]\n",
"print 'E=\\n',E\n",
"F=np.identity(3)\n",
"F[2,:]=F[2,:]+F[0,:]\n",
"print 'F=\\n',F\n",
"G=np.identity(3)\n",
"G[2,:]=G[2,:]+G[1,:]\n",
"print 'G=\\n',G\n",
"print 'GE=',G.dot(E)\n",
"print 'EG=\\n',E.dot(G)\n",
"print 'Here EG is not equal to GE,Therefore these two matrices do not commute and shows that most matrices do not commute.'\n",
"print 'GFE=\\n',G.dot(F.dot(E))\n",
"print 'EFG=\\n',E.dot(F.dot(G))\n",
"print 'The product GFE is the true order of elimation.It is the matrix that takes the original A to the upper triangular U.'"
]
}
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