{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "\n",
      "import numpy as np\n",
      "import numpy.linalg as la"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = 5\n",
      "\n",
      "np.random.seed(70)\n",
      "eigvecs = np.random.randn(n, n)\n",
      "eigvals = np.sort(np.random.randn(n))\n",
      "\n",
      "A = np.dot(la.solve(eigvecs, np.diag(eigvals)), eigvecs)\n",
      "print eigvals"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[-1.3657822  -0.78460489 -0.08829521  0.30824369  0.52110266]\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = np.random.randn(n, n)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Run this cell in-place (Ctrl-Enter) many times.\n",
      "\n",
      "Q, R = la.qr(X)\n",
      "X = np.dot(A, Q)\n",
      "print Q"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[-0.36366705 -0.24462159  0.39052837  0.06435026 -0.8070026 ]\n",
        " [-0.42943052 -0.63956476  0.08336455 -0.49910706  0.3879289 ]\n",
        " [-0.040959   -0.42224061 -0.83670032  0.25124759 -0.23841653]\n",
        " [ 0.80079022 -0.35639657  0.08781549 -0.40630442 -0.24273785]\n",
        " [-0.20098031  0.47519633 -0.36436102 -0.72009898 -0.28721745]]\n"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "la.norm(\n",
      "    np.dot(np.dot(Q, R), Q.T)\n",
      "    - A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 52,
       "text": [
        "3.1730296900420734e-08"
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.set_printoptions(linewidth=100)\n",
      "np.dot(np.dot(Q.T, A), Q)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 53,
       "text": [
        "array([[ -1.36578220e+00,   8.74082014e-02,   3.03787100e+00,  -2.62444769e-01,  -2.97605237e-01],\n",
        "       [  1.62514446e-12,  -7.84604888e-01,  -4.58332378e-01,  -1.38868012e+00,   2.45153479e-01],\n",
        "       [  1.90819582e-16,   5.14360649e-09,   5.21102659e-01,  -1.52024449e-01,   7.37881218e-02],\n",
        "       [  6.24500451e-17,   2.22044605e-16,  -2.84249013e-12,   3.08243691e-01,   1.76857006e-01],\n",
        "       [ -3.70363462e-16,   5.72458747e-17,   4.12864187e-16,   2.46330734e-16,  -8.82952134e-02]])"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}