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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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     "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": [
    "import numpy as np\n",
    "import sympy as sp\n",
    "A = sp.Matrix(([5, -6, -6],[-1, 4 ,2],[3, -6, -4]))\n",
    "print 'A = \\n',A\n",
    "x=sp.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 = np.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'"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Page 247 Example 7.6"
   ]
  },
  {
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   "execution_count": 2,
   "metadata": {
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   "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": [
    "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 = sp.Matrix(([2, 0, 0],[a, 2, 0],[b, c, -1]))\n",
    "print 'A = \\n',A\n",
    "print 'Characteristic polynomial for A is:'\n",
    "x=sp.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 = sp.Matrix(([2, 0, 0],[a, 2, 0],[b, c, -1]))\n",
    "print 'A = \\n',A\n",
    "print 'Characteristic polynomial for A is:'\n",
    "x=sp.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": 3,
   "metadata": {
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   },
   "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": [
    "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 = sp.Matrix(([2, 0 ,0 ,0],[1, 2, 0, 0],[0, 0 ,2 ,0],[0, 0, 1, 2]))\n",
    "x=sp.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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