{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load_ext sympy.interactive.ipythonprinting\n",
      "\n",
      "import sympy as sp"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "/usr/lib/python2.7/dist-packages/IPython/frontend.py:30: UserWarning: The top-level `frontend` package has been deprecated. All its subpackages have been moved to the top `IPython` level.\n",
        "  warn(\"The top-level `frontend` package has been deprecated. \"\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eps = sp.Symbol(\"epsilon\")\n",
      "lam = sp.Symbol(\"lambda\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "m = sp.Matrix([[1, eps], [eps, 1]])\n",
      "m"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{smallmatrix}1 & \\epsilon\\\\\\epsilon & 1\\end{smallmatrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACUAAAAXBAMAAAB+ApuWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEu5mrVO8i\nEGbo53ZYAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjklEQVQYGWNgAAGWAjCFTLB0KiBzIWwuVDEh\nE6AwkhifUQJDGEglkhgDK6qYRyBQHlWMTRekB1WMuTQEQ4zfAUXdBhCP8wGSGE/TTwEgl7dREEjC\nzAPJwwCaGNjbqGIQb6OKQbxDqhjYo9CggOqFeBRVDOJRVDGIR1HFIB4FioG8DTRPSAXmUQawtxmV\nEgCmsx7UPBtlQwAAAABJRU5ErkJggg==\n",
       "prompt_number": 10,
       "text": [
        "\u23a11  \u03b5\u23a4\n",
        "\u23a2    \u23a5\n",
        "\u23a3\u03b5  1\u23a6"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "m.charpoly(lam)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{PurePoly}{\\left( - \\epsilon^{2} + \\lambda^{2} - 2 \\lambda + 1, \\lambda, domain=\\mathbb{Z}\\left[\\epsilon\\right] \\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAbEAAAAcBAMAAAAdC5wGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 11,
       "text": [
        "PurePoly(lambda**2 - 2*lambda - epsilon**2 + 1, lambda, domain='ZZ[epsilon]')"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Observe the occurrence of $(1-\\epsilon^2)$ above."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}