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diff --git a/sample_notebooks/Vaibhav Vajani/chapter2_1.ipynb b/sample_notebooks/Vaibhav Vajani/chapter2_1.ipynb new file mode 100644 index 00000000..40066b00 --- /dev/null +++ b/sample_notebooks/Vaibhav Vajani/chapter2_1.ipynb @@ -0,0 +1,1360 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 2: Determinants and Matrices" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.1, page no. 55" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Determinant of A is: a*b*c - a*f**2 - b*g**2 - c*h**2 + 2*f*g*h\n" + ] + } + ], + "source": [ + "import sympy\n", + "\n", + "a = sympy.Symbol('a')\n", + "h = sympy.Symbol('h')\n", + "g = sympy.Symbol('g')\n", + "b = sympy.Symbol('b')\n", + "f = sympy.Symbol('f')\n", + "c = sympy.Symbol('c')\n", + "A = sympy.Matrix([[a,h,g],[h,b,f],[g,f,c]])\n", + "\n", + "print \"Determinant of A is: \", A.det()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.2, page no. 55" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false, + "scrolled": true + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Determinanat of a is: 88.0\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.array([[0,1,2,3],[1,0,3,0],[2,3,0,1],[3,0,1,2]])\n", + "print \"Determinant of a is:\",numpy.linalg.det(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.3, page no. 56" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "-a**3*b**2*c + a**3*b*c**2 + a**2*b**3*c - a**2*b*c**3 + a**2*b - a**2*c - a*b**3*c**2 + a*b**2*c**3 - a*b**2 + a*c**2 + b**2*c - b*c**2\n" + ] + } + ], + "source": [ + "import numpy\n", + "import sympy\n", + "\n", + "a = sympy.Symbol('a');\n", + "b = sympy.Symbol('b');\n", + "c = sympy.Symbol('c');\n", + "A = sympy.Matrix([[a,a**2,a**3-1],[b,b**2,b**3-1],[c,c**2,c**3-1]])\n", + "\n", + "print A.det()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.4, page no. 57" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Determinanat of a is: -24.0\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.array([[21,17,7,10],[24,22,6,10],[6,8,2,3],[6,7,1,2]])\n", + "print \"Determinanat of a is:\",numpy.linalg.det(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "imp" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "a: x**y*log(x)\n", + "b: x**y*y*log(x)/x + x**y/x\n", + "c: x**y*y**2*log(x)/x**2 - x**y*y*log(x)/x**2 + 2*x**y*y/x**2 - x**y/x**2\n", + "d: x**y*y/x\n", + "e: x**y*y*log(x)/x + x**y/x\n", + "f: x**y*y**2*log(x)/x**2 - x**y*y*log(x)/x**2 + 2*x**y*y/x**2 - x**y/x**2\n", + "Clearly c = f\n" + ] + } + ], + "source": [ + "import sympy\n", + "\n", + "x = sympy.Symbol('x');\n", + "y = sympy.Symbol('y')\n", + "u = x**y\n", + "a = sympy.diff(u, y)\n", + "b = sympy.diff(a, x)\n", + "c = sympy.diff(b, x)\n", + "d = sympy.diff(u, x)\n", + "e = sympy.diff(d, y)\n", + "f = sympy.diff(e, x)\n", + "print \"a: \", a\n", + "print \"b: \", b\n", + "print \"c: \", c\n", + "print \"d: \", d\n", + "print \"e: \", e\n", + "print \"f: \", f\n", + "print \"Clearly c = f\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.17, page no. 65" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A*B=\n", + "[[ 5 9 13]\n", + " [-1 2 4]\n", + " [-2 2 4]]\n", + "\n", + "B*A=\n", + "[[-1 12 11]\n", + " [-1 7 8]\n", + " [-2 -1 5]]\n", + "Clearly AB is not equal to BA\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.array([[1,3,0],[-1,2,1],[0,0,2]])\n", + "B = numpy.array([[2,3,4],[1,2,3],[-1,1,2]])\n", + "ma = numpy.matrix(A)\n", + "mb = numpy.matrix(B)\n", + "print \"A*B=\"\n", + "print ma*mb\n", + "print \"\"\n", + "print \"B*A=\"\n", + "print mb*ma\n", + "print \"Clearly AB is not equal to BA\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.18, page no. 65" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "AB=C−−>B=inv(A)∗C\n", + "\n", + "[[ 1.00000000e+00 -1.77635684e-15 -8.88178420e-16]\n", + " [ -2.22044605e-16 2.00000000e+00 -1.11022302e-16]\n", + " [ 1.77635684e-15 0.00000000e+00 1.00000000e+00]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[3,2,2],[1,3,1],[5,3,4]])\n", + "C = numpy.matrix([[3,4,2],[1,6,1],[5,6,4]])\n", + "print \"AB=C−−>B=inv(A)∗C\"\n", + "print \"\"\n", + "B = numpy.linalg.inv(A)*C \n", + "print B" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.19, page no. 66" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Aˆ3−4∗Aˆ2−3A+11*I=\n", + "\n", + "[[ 0. 0. 0.]\n", + " [ 0. 0. 0.]\n", + " [ 0. 0. 0.]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.array([[1,3,2],[2,0,-1],[1,2,3]])\n", + "I = numpy.eye(3)\n", + "A = numpy.matrix(A)\n", + "print \"Aˆ3−4∗Aˆ2−3A+11*I=\"\n", + "print \"\"\n", + "print A**3-4*A**2-3*A+11*I" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.20, page no. 67" + ] + }, + { + "cell_type": "code", + "execution_count": 58, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Enter the value of n: 3\n", + "Calculating A ^ n: \n", + "[[ 31 -75]\n", + " [ 12 -29]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "0\n", + "A = numpy.matrix([[11,-25],[4, -9]])\n", + "n = int(raw_input(\"Enter the value of n: \"))\n", + "print \"Calculating A ^ n: \"\n", + "print A**n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.23, page no. 70" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Inverse of A is: \n", + "[[ 3. 1. 1.5 ]\n", + " [-1.25 -0.25 -0.75]\n", + " [-0.25 -0.25 -0.25]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[1,1,3],[1,3,-3],[-2,-4,-4]])\n", + "print \"Inverse of A is: \"\n", + "print numpy.linalg.inv(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.24.1, page no. 71" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Rank of A is: 2\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[1,2,3],[1,4,2],[2,6,5]])\n", + "print \"Rank of A is:\",numpy.linalg.matrix_rank(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.24.2, page no. 71" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Rank of A is: 2\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[0,1,-3,-1],[1,0,1,1],[3,1,0,2],[1,1,-2,0]])\n", + "print \"Rank of A is:\",numpy.linalg.matrix_rank(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.25, page no. 72" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Inverse of A is: \n", + "[[ 3. 1. 1.5 ]\n", + " [-1.25 -0.25 -0.75]\n", + " [-0.25 -0.25 -0.25]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[1,1,3],[1,3,-3],[-2,-4,-4]])\n", + "print \"Inverse of A is: \"\n", + "print numpy.linalg.inv(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.26, page no. 73" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Rank of A is: 3\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[2,3,-1,-1],[1,-1,-2,-4],[3,1,3,-2],[6,3,0,-7]])\n", + "r,p = numpy.linalg.eigh ( A )\n", + "print \"Rank of A is:\",numpy.linalg.matrix_rank(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.28, page no. 75" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Inverse of A is: \n", + "[[ 1.4 0.2 -0.4]\n", + " [-1.5 0. 0.5]\n", + " [ 1.1 -0.2 -0.1]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[1,1,1],[4,3,-1],[3,5,3]])\n", + "print \"Inverse of A is: \"\n", + "print numpy.linalg.inv(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.31, page no. 78" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " The equations can be rewritten as AX=B where X=[ x1 ; x2 ; x3 ; x4 ] and \n", + "Determinant of A=\n", + "8.0\n", + "Inverse of A =\n", + "[[ 0.5 0.5 0.5 -0.5]\n", + " [-0.5 0. 0. 0.5]\n", + " [ 0. -0.5 0. 0.5]\n", + " [ 0. 0. -0.5 0.5]]\n", + "X= [[ 1.]\n", + " [-1.]\n", + " [ 2.]\n", + " [-2.]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "print \"The equations can be rewritten as AX=B where X=[ x1 ; x2 ; x3 ; x4 ] and \"\n", + "A = numpy.matrix([[1,-1,1,1],[1,1,-1,1],[1,1,1,-1],[1,1,1,1]])\n", + "B = numpy.matrix([[2],[-4],[4],[0]])\n", + "print \"Determinant of A=\"\n", + "print numpy.linalg.det(A)\n", + "print \"Inverse of A =\"\n", + "print numpy.linalg.inv(A)\n", + "print \"X=\",numpy.linalg.inv(A)*B" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.32, page no. 78" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The equations can be rewritten as AX=B where X=[x;y;z] and\n", + "Determinant of A=\n", + "-8.79296635503e-14\n", + "Since det(A)=0 , hence, this system of equation will have infinite solutions.. hence, the system is consistent\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "print \"The equations can be rewritten as AX=B where X=[x;y;z] and\"\n", + "A = numpy.matrix([[5,3,7],[3,26,2],[7,2,10]])\n", + "B = numpy.matrix([[4],[9],[5]])\n", + "print \"Determinant of A=\"\n", + "print numpy.linalg.det(A)\n", + "print \"Since det(A)=0 , hence, this system of equation will have infinite solutions.. hence, the system is consistent\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.34.1, page no. 80" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Rank of A is 3\n", + "Equations have only a trivial solution : x=y=z=0\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[1,2,3],[3,4,4],[7,10,12]])\n", + "p = numpy.linalg.matrix_rank(A)\n", + "print \"Rank of A is\",p\n", + "if p==3:\n", + " print \"Equations have only a trivial solution : x=y=z=0\"\n", + "else:\n", + " print \"Equations have infinite no . of solutions.\"" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Example 2.34.2, page no. 80" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Rank of A is 2\n", + "Equations have infinite no. of solutions.\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[4,2,1,3],[6,3,4,7],[2,1,0,1]])\n", + "p = numpy.linalg.matrix_rank(A)\n", + "print \"Rank of A is\",p\n", + "if p ==4:\n", + " print \"Equations have only a trivial solution : x=y=z=0\"\n", + "else:\n", + " print \"Equations have infinite no. of solutions.\"" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Example 2.38, page no. 83" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The given equations can be written as Y=AX where\n", + "Determinant of A is -1.0\n", + "Since, its non−singular, hence transformation is regular\n", + "Inverse of A is\n", + "[[ 2. -2. -1.]\n", + " [-4. 5. 3.]\n", + " [ 1. -1. -1.]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "print \"The given equations can be written as Y=AX where\"\n", + "A = numpy.matrix([[2,1,1],[1,1,2],[1,0,-2]])\n", + "print \"Determinant of A is\",numpy.linalg.det ( A )\n", + "print \"Since, its non−singular, hence transformation is regular\"\n", + "print\"Inverse of A is\"\n", + "print numpy.linalg.inv ( A )" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.39, page no. 84" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[-0.66666667 0.33333333 0.66666667]\n", + " [ 0.66666667 0.66666667 0.33333333]\n", + " [ 0.33333333 -0.66666667 0.66666667]]\n", + "A transpose is equal to\n", + "[[-0.66666667 0.66666667 0.33333333]\n", + " [ 0.33333333 0.66666667 -0.66666667]\n", + " [ 0.66666667 0.33333333 0.66666667]]\n", + "A∗(transpose of A)=\n", + "[[ 1. 0. 0.]\n", + " [ 0. 1. 0.]\n", + " [ 0. 0. 1.]]\n", + "Hence, A is orthogonal\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[-2./3,1./3,2./3],[2./3,2./3,1./3],[1./3,-2./3,2./3]])\n", + "print A\n", + "print \"A transpose is equal to\"\n", + "print A.transpose()\n", + "print \"A∗(transpose of A)=\"\n", + "print A*A.transpose()\n", + "print \"Hence, A is orthogonal\"" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Example 2.42, page no. 87" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Let R represents the matrix of transformation and P represents a diagonal matrix whose values are the eigenvalues of A. then\n", + "R is normalised. let U represents unnormalised version of r\n", + "Two eigen vectors are the two columns of U\n", + "[[ 4. 1.]\n", + " [ 0. 0.]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "import math\n", + "\n", + "A = numpy.matrix([[5,4],[1,2]])\n", + "print \"Let R represents the matrix of transformation and P represents a diagonal matrix whose values are the eigenvalues of A. then\"\n", + "P,R= numpy.linalg.eig(A)\n", + "U = numpy.zeros([2, 2])\n", + "print \"R is normalised. let U represents unnormalised version of r\"\n", + "U[0,0]= R[0,0]*math.sqrt(17)\n", + "U[0,1]= R[0,1]*math.sqrt(17)\n", + "U[0,1]= R[1,1]*math.sqrt(2)\n", + "print \"Two eigen vectors are the two columns of U\"\n", + "print U" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Examle 2.43, page no. 88" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Let Rrepresents the matrix of transformation and Prepresents a diagonalmatrix whose values are the eigenvalues of A. then\n", + "R is normalised. let U represents unnormalised version of r\n", + "[-2. 3. 6.]\n", + "Three eigen vectors are the three columns of U\n", + "[[ -2.82842712 5.19615242 14.69693846]\n", + " [ 0. 0. 0. ]\n", + " [ 0. 0. 0. ]]\n" + ] + } + ], + "source": [ + "import numpy,math\n", + "\n", + "A = numpy.matrix([[1,1,3],[1,5,1],[3,1,1]])\n", + "U = numpy.zeros([3,3])\n", + "print \"Let Rrepresents the matrix of transformation and Prepresents a diagonalmatrix whose values are the eigenvalues of A. then\"\n", + "R,P = numpy.linalg.eig(A)\n", + "print \"R is normalised. let U represents unnormalised version of r\"\n", + "print R\n", + "U[0,0] = R[0]*math.sqrt(2) \n", + "U[0,1] = R[1]*math.sqrt(3)\n", + "U[0,2] = R[2]*math.sqrt(6)\n", + "print \"Three eigen vectors are the three columns of U\"\n", + "print U" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.44, page no. 89" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Let Rrepresents the matrix of transformation and Prepresents a diagonalmatrix whose values are the eigenvalues of A. then\n", + "R is normalised. let U represents unnormalised version of r\n", + "[ 3. 2. 5.]\n", + "Three eigen vectors are the three columns of U\n", + "[[ 3. 2. 18.70828693]\n", + " [ 0. 0. 0. ]\n", + " [ 0. 0. 0. ]]\n" + ] + } + ], + "source": [ + "import numpy,math\n", + "\n", + "A = numpy.matrix([[3,1,4],[0,2,6],[0,0,5]])\n", + "U = numpy.zeros([3,3])\n", + "print \"Let Rrepresents the matrix of transformation and Prepresents a diagonalmatrix whose values are the eigenvalues of A. then\"\n", + "R,P = numpy.linalg.eig(A)\n", + "print \"R is normalised. let U represents unnormalised version of r\"\n", + "print R\n", + "U[0,0] = R[0]*math.sqrt(1) \n", + "U[0,1] = R[1]*math.sqrt(1)\n", + "U[0,2] = R[2]*math.sqrt(14)\n", + "print \"Three eigen vectors are the three columns of U\"\n", + "print U" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.45, page no. 90" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Eigen values of A are\n", + "(array([-1., 5.]), matrix([[-0.89442719, -0.70710678],\n", + " [ 0.4472136 , -0.70710678]]))\n", + "Let\n", + "Hence, the characteristic equation is ( x−a ) ( x−b)\n", + "[-8 -5]\n", + "Aˆ2−4∗A−5∗ I=\n", + "[[ 0. 0.]\n", + " [ 0. 0.]]\n", + "Inverse of A=\n", + "[[-0.6 0.8]\n", + " [ 0.4 -0.2]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "x = numpy.poly([0])\n", + "A = numpy.matrix([[1,4],[2,3]])\n", + "I = numpy.eye(2)\n", + "print \"Eigen values of A are\"\n", + "print numpy.linalg.eig(A)\n", + "print \"Let\"\n", + "a = -1;\n", + "b = 5;\n", + "print \"Hence, the characteristic equation is ( x−a ) ( x−b)\"\n", + "print ( x - a ) *( x - b )\n", + "\n", + "print \"Aˆ2−4∗A−5∗ I=\"\n", + "print A**2-4*A-5* I\n", + "print \"Inverse of A=\"\n", + "print numpy.linalg.inv ( A )" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.46, page no. 91" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Egenvalues of A are\n", + "(array([ 4.25683813, 0.40327935, -4.66011748]), matrix([[ 0.10296232, -0.91299477, -0.48509974],\n", + " [-0.90473047, 0.40531299, 0.37306899],\n", + " [ 0.41335402, 0.04649661, 0.79088417]]))\n", + "Let\n", + "Hence, the characteristic equation is ( x−a ) ( x−b) ( x−c )\n", + "[-10.99999905 8.00000095]\n", + "Inverse of A=\n", + "[[ 3. 1. 1.5 ]\n", + " [-1.25 -0.25 -0.75]\n", + " [-0.25 -0.25 -0.25]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "x = numpy.poly([0])\n", + "A = numpy.matrix([[1,1,3],[1,3,-3],[-2,-4,-4]])\n", + "print \"Egenvalues of A are\"\n", + "print numpy.linalg.eig(A)\n", + "print \"Let\"\n", + "a =4.2568381\n", + "b =0.4032794\n", + "c = -4.6601175\n", + "print \"Hence, the characteristic equation is ( x−a ) ( x−b) ( x−c )\"\n", + "p = (x-a)*(x-b)*(x-c)\n", + "print p\n", + "print \"Inverse of A=\"\n", + "print numpy.linalg.inv(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.47, page no. 91" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Eigenvalues of A are\n", + "(array([ 3., 1., 1.]), matrix([[ 0.70710678, -0.70710678, -0.40824829],\n", + " [ 0. , 0. , 0.81649658],\n", + " [ 0.70710678, 0.70710678, -0.40824829]]))\n", + "Let\n", + "Hence, the characteristic equation is (x−a)(x−b)(x−c)=\n", + "[ 0 -3]\n", + "Aˆ8−5∗Aˆ7+7∗Aˆ6−3∗Aˆ5+Aˆ4−5∗Aˆ3+8∗Aˆ2−2∗A+I =\n", + "[[ 8. 5. 5.]\n", + " [ 0. 3. 0.]\n", + " [ 5. 5. 8.]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "x = numpy.poly([0])\n", + "A = numpy.matrix([[2,1,1],[0,1,0],[1,1,2]])\n", + "I = numpy.eye(3)\n", + "print \"Eigenvalues of A are\"\n", + "print numpy.linalg.eig(A)\n", + "print \"Let\"\n", + "a =1\n", + "b =1\n", + "c =3\n", + "print \"Hence, the characteristic equation is (x−a)(x−b)(x−c)=\"\n", + "p = (x-a)*(x-b)*(x-c)\n", + "print p\n", + "print \"Aˆ8−5∗Aˆ7+7∗Aˆ6−3∗Aˆ5+Aˆ4−5∗Aˆ3+8∗Aˆ2−2∗A+I =\"\n", + "print A**8-5*A**7+7*A**6-3*A**5+A**4-5*A**3+8*A**2-2*A+I" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "## Example 2.48, page no. 93" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "R is matrix of transformation and D is a diagonal matrix\n", + "[-1.65544238 -0.21075588 2.86619826]\n", + "[[-0.87936655 -0.34661859 -0.32645063]\n", + " [ 0.11410244 0.51222983 -0.85123513]\n", + " [-0.46227167 0.78579651 0.41088775]]\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[-1,2,-2],[1,2,1],[-1,-1,0]])\n", + "print \"R is matrix of transformation and D is a diagonal matrix\"\n", + "[R,D]= numpy.linalg.eigh(A)\n", + "print R\n", + "print D" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.49, page no. 93" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "R is matrix of transformation and D is a diagonal matrix\n", + "R is normalised, let P denotes unnormalised version of R . Then \n", + "[ 3. 2. 5.]\n", + "[[ 4.24264069 3.46410162 12.24744871]\n", + " [ 0. 0. 0. ]\n", + " [ 0. 0. 0. ]]\n", + "A^4= [[ 81 65 1502]\n", + " [ 0 16 1218]\n", + " [ 0 0 625]]\n" + ] + } + ], + "source": [ + "import numpy,math\n", + "\n", + "A = numpy.matrix([[3,1,4],[0,2,6],[0,0,5]])\n", + "P = numpy.zeros([3,3])\n", + "print \"R is matrix of transformation and D is a diagonal matrix\"\n", + "R,D = numpy.linalg.eig(A)\n", + "print \"R is normalised, let P denotes unnormalised version of R . Then \"\n", + "print R\n", + "P[0,0] = R[0]*math.sqrt(2) \n", + "P[0,1] = R[1]*math.sqrt(3)\n", + "P[0,2] = R[2]*math.sqrt(6)\n", + "print P\n", + "print \"A^4= \",A**4" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.50, page no. 94" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "3∗xˆ2+5∗yˆ2+3∗zˆ2−2∗y∗z+2∗z∗x−2∗x∗y\n", + "The matrix of the given quadratic form is\n", + "Let R represents the matrix of transformation and Prepresents a diagonal matrix whose values are the eigenvalues of A. then\n", + "So, canonical form is 2∗xˆ2+3∗yˆ2+6∗zˆ2\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "print \"3∗xˆ2+5∗yˆ2+3∗zˆ2−2∗y∗z+2∗z∗x−2∗x∗y\"\n", + "print \"The matrix of the given quadratic form is\"\n", + "A = numpy.matrix([[3,-1,1],[-1,5,-1],[1,-1,3]])\n", + "print \"Let R represents the matrix of transformation and Prepresents a diagonal matrix whose values are the eigenvalues of A. then\"\n", + "[R,P] = numpy.linalg.eig(A)\n", + "print \"So, canonical form is 2∗xˆ2+3∗yˆ2+6∗zˆ2\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.51, page no. 95" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "2∗x1∗x2+2∗x1∗x3−2∗x2∗x3\n", + "The matrix of the given quadratic form is\n", + "Let R represents the matrix of transformation and P represents a diagonal matrix whose values are the eigenvalues of A. then\n", + "so, canonical form is −2∗xˆ2+yˆ2+ zˆ2\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "print \"2∗x1∗x2+2∗x1∗x3−2∗x2∗x3\"\n", + "print \"The matrix of the given quadratic form is\"\n", + "A = numpy.matrix([[0,1,1],[1,0,-1],[1,-1,0]])\n", + "print \"Let R represents the matrix of transformation and P represents a diagonal matrix whose values are the eigenvalues of A. then\"\n", + "[R,P] = numpy.linalg.eig(A)\n", + "print \"so, canonical form is −2∗xˆ2+yˆ2+ zˆ2\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.52, page no. 96" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A∗= [[ 2.-1.j -5.-0.j]\n", + " [ 3.-0.j 0.-1.j]\n", + " [-1.-3.j 4.+2.j]]\n", + "AA∗= [[ 24.+0.j -20.+2.j]\n", + " [-20.-2.j 46.+0.j]]\n", + "Clearly, AA∗ is hermitian matrix\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "'''\n", + "A=[2+%i 3 -1+3*%i;-5 %i 4-2*%i]\n", + "'''\n", + "\n", + "A = numpy.matrix([[2+1j,3,-1+3*1j],[-5,1j,4-2*1j]])\n", + "#A = A.getH()\n", + "print \"A∗=\", A.getH()\n", + "print \"AA∗=\", A*(A.getH())\n", + "print \"Clearly, AA∗ is hermitian matrix\"" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.53, page no. 97" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " A∗= [[ 0.-0.j 0.-0.j]\n", + " [-0.-0.j 0.-0.j]]\n", + "AA∗= [[ 0.+0.j 0.+0.j]\n", + " [ 0.+0.j 0.+0.j]]\n", + "A∗A= [[ 0.+0.j 0.+0.j]\n", + " [ 0.+0.j 0.+0.j]]\n" + ] + } + ], + "source": [ + "import numpy \n", + "\n", + "A = numpy.matrix([[(1/2)*(1+1j),(1/2)*(-1+1j)],[(1/2)*(1+1j),(1/2)*(1-1j)]])\n", + "print \"A∗=\", A.getH()\n", + "print \"AA∗=\", A*(A.getH())\n", + "print \"A∗A=\", (A.getH())*A" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2.54, page no. 97" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "I−A=\n", + "inverse of (I+A)=\n", + "[[ 0.16666667+0.j -0.16666667-0.33333333j]\n", + " [ 0.16666667-0.33333333j 0.16666667+0.j ]]\n", + "((I−A)(inverse(I+A)))∗((I−A)(inverse(I+A)))=\n", + "[[ 1.11111111e-01-0.44444444j -2.77555756e-17+0.88888889j]\n", + " [ -2.77555756e-17+0.88888889j 1.11111111e-01+0.44444444j]]\n", + "((I−A)(inverse(I+A)))((I−A)(inverse(I+A)))∗=\n", + "[[ 1.11111111e-01+0.44444444j 1.11022302e-16+0.88888889j]\n", + " [ 1.11022302e-16+0.88888889j 1.11111111e-01-0.44444444j]]\n", + "Clearly, the product is an identity matrix.hence, it is a unitary matrix\n" + ] + } + ], + "source": [ + "import numpy\n", + "\n", + "A = numpy.matrix([[0,1+2*1j],[-1+2*1j,0]])\n", + "I = numpy.eye(2)\n", + "print \"I−A=\"\n", + "I-A\n", + "print \"inverse of (I+A)=\"\n", + "print numpy.linalg.inv(I+A)\n", + "print \"((I−A)(inverse(I+A)))∗((I−A)(inverse(I+A)))=\"\n", + "print (((I-A)*(numpy.linalg.inv(I+A))).T)*((I-A)*(numpy.linalg.inv(I+A)))\n", + "print \"((I−A)(inverse(I+A)))((I−A)(inverse(I+A)))∗=\"\n", + "print ((I-A)*(numpy.linalg.inv(I+A)))*(((I-A)*(numpy.linalg.inv(I+A))).T)\n", + "print \"Clearly, the product is an identity matrix.hence, it is a unitary matrix\"" + ] + } + ], + "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": "2.7.11+" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} |