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{
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{
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"source": [
"# Chapter 7 - The rational and jordan forms"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 239 Example 7.3"
]
},
{
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"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \n",
"Matrix([[5, -6, -6], [-1, 4, 2], [3, -6, -4]])\n",
"Characteristic polynomial for linear operator T on R**3 will be:\n",
"f = x**3 - 5*x**2 + 8*x - 4\n",
"or\n",
"(x-1)(x-2)**2\n",
"The minimal polynomial for T is:\n",
"p = (x - 2)*(x - 1)\n",
"or\n",
"p = (x-1)(x-2)\n",
"So in cyclic decomposition of T, a1 will have p as its T-annihilator.\n",
"Another vector a2 that generate cyclic subspace of dimension 1 will have its T-annihilator as p2.\n",
"p2 = x - 2\n",
"pp2 = (x - 2)**2*(x - 1)\n",
"i.e., pp2 = f\n",
"Therefore, A is similar to B\n",
"B = \n",
"[[ 0 -2 0]\n",
" [ 1 3 0]\n",
" [ 0 0 2]]\n",
"Thus, we can see thet Matrix of T in ordered basis is B\n"
]
}
],
"source": [
"from numpy import array\n",
"from sympy import Symbol,Matrix\n",
"A = Matrix(([5, -6, -6],[-1, 4 ,2],[3, -6, -4]))\n",
"print 'A = \\n',A\n",
"x=Symbol('x')\n",
"f = A.charpoly(x).as_expr()\n",
"print 'Characteristic polynomial for linear operator T on R**3 will be:'\n",
"print 'f = ',f\n",
"print 'or'\n",
"print '(x-1)(x-2)**2'\n",
"print 'The minimal polynomial for T is:'\n",
"p = (x-1)*(x-2)#\n",
"print 'p = ',p\n",
"print 'or'\n",
"print 'p = (x-1)(x-2)'\n",
"print 'So in cyclic decomposition of T, a1 will have p as its T-annihilator.'\n",
"print 'Another vector a2 that generate cyclic subspace of dimension 1 will have its T-annihilator as p2.'\n",
"p2 = x-2#\n",
"print 'p2 = ',p2\n",
"print 'pp2 = ',p*p2\n",
"print 'i.e., pp2 = f'\n",
"print 'Therefore, A is similar to B'\n",
"B = array([[0, -2, 0],[1, 3, 0],[0, 0 ,2]])\n",
"print 'B = \\n',B\n",
"print 'Thus, we can see thet Matrix of T in ordered basis is B'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 247 Example 7.6"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \n",
"2 0 0\n",
"a 2 0\n",
"b c -1\n",
"A = \n",
"Matrix([[2, 0, 0], [1, 2, 0], [0, 0, -1]])\n",
"Characteristic polynomial for A is:\n",
"p = x**3 - 3*x**2 + 4\n",
"In this case, minimal polynomial is same as characteristic polynomial.\n",
"-----------------------------------------\n",
"A = \n",
"Matrix([[2, 0, 0], [0, 2, 0], [0, 0, -1]])\n",
"Characteristic polynomial for A is:\n",
"p = x**3 - 3*x**2 + 4\n",
"In this case, minimal polynomial is: (x-2)(x+1)\n",
"or\n",
"(x - 2)*(x + 1)\n",
"(A-2I)(A+I) = \n",
"0 0 0\n",
"3a 0 0\n",
"ac 0 0\n",
"if a = 0, A is similar to diagonal matrix.\n"
]
}
],
"source": [
"from numpy import array\n",
"from sympy import Symbol,Matrix\n",
"\n",
"print 'A = '\n",
"print '2 0 0'\n",
"print 'a 2 0'\n",
"print 'b c -1'\n",
"a = 1#\n",
"b = 0#\n",
"c = 0#\n",
"A = Matrix(([2, 0, 0],[a, 2, 0],[b, c, -1]))\n",
"print 'A = \\n',A\n",
"print 'Characteristic polynomial for A is:'\n",
"x=Symbol('x')\n",
"print 'p = ',A.charpoly(x).as_expr()\n",
"print 'In this case, minimal polynomial is same as characteristic polynomial.'\n",
"print '-----------------------------------------'\n",
"a = 0#\n",
"b = 0#\n",
"c = 0#\n",
"A = Matrix(([2, 0, 0],[a, 2, 0],[b, c, -1]))\n",
"print 'A = \\n',A\n",
"print 'Characteristic polynomial for A is:'\n",
"x=Symbol('x')\n",
"print 'p = ',A.charpoly(x).as_expr()\n",
"print 'In this case, minimal polynomial is:',\n",
"print '(x-2)(x+1)'\n",
"print 'or'\n",
"s = (x-2)*(x+1)#\n",
"print s\n",
"print '(A-2I)(A+I) = '\n",
"print '0 0 0'\n",
"print '3a 0 0'\n",
"print 'ac 0 0'\n",
"print 'if a = 0, A is similar to diagonal matrix.'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Page 247 Example 7.7"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A = \n",
"2 0 0 0\n",
"1 2 0 0\n",
"0 0 2 0\n",
"0 0 a 2\n",
"Considering a = 1\n",
"Characteristic polynomial for A is:\n",
"p = x**4 - 8*x**3 + 24*x**2 - 32*x + 16\n",
"or\n",
"(x-2)**4\n",
"Minimal polynomial for A =\n",
"(x-2)**2\n",
"For a = 0 and a = 1, characteristic and minimal polynomial are same.\n",
"But for a=0, the solution space of (A - 2I) has 3 dimension whereas for a = 1, it has 2 dimension. \n"
]
}
],
"source": [
"from numpy import array\n",
"from sympy import Symbol,Matrix\n",
"print 'A = '\n",
"print '2 0 0 0'\n",
"print '1 2 0 0'\n",
"print '0 0 2 0'\n",
"print '0 0 a 2'\n",
"print 'Considering a = 1'\n",
"A = Matrix(([2, 0 ,0 ,0],[1, 2, 0, 0],[0, 0 ,2 ,0],[0, 0, 1, 2]))\n",
"x=Symbol('x')\n",
"p = A.charpoly(x).as_expr()\n",
"print 'Characteristic polynomial for A is:'\n",
"print 'p = ',p\n",
"print 'or'\n",
"print '(x-2)**4'\n",
"print 'Minimal polynomial for A ='\n",
"print '(x-2)**2'\n",
"print 'For a = 0 and a = 1, characteristic and minimal polynomial are same.'\n",
"print 'But for a=0, the solution space of (A - 2I) has 3 dimension whereas for a = 1, it has 2 dimension. '"
]
}
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