Evaluation Task Machine Learning for Turbulent Fluid Dynamics.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generating the Symmetric Matrices\n",
    "To generate symmetric matrix from any general matrix M, we can do any one of the following\n",
    "\n",
    "M_symm = (M+M.T)/2\n",
    "\n",
    "M_symm = M*M.T\n",
    "\n",
    "Using rules of matrix transpose, it can be shown that these two matrices are symmetric for any matrix M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.51362694  0.02375238 -0.57354865 -0.52947567 -0.56655558 -0.00100043\n",
      "   0.21929146 -0.1587675   0.64460971 -0.22849717]\n",
      " [ 0.02375238  0.85687269  0.86637343 -0.6393855   0.50356679 -0.56710161\n",
      "  -0.28202469  0.25638067  0.74370076  0.39420788]\n",
      " [-0.57354865  0.86637343 -0.20362728  0.41619781 -0.0674289   0.73669033\n",
      "  -0.6785701   0.14691901  0.02618289  0.10853407]\n",
      " [-0.52947567 -0.6393855   0.41619781 -0.61600671 -0.12818749  0.04868083\n",
      "  -0.57794609  0.33673342 -0.06714558  0.05619415]\n",
      " [-0.56655558  0.50356679 -0.0674289  -0.12818749 -0.58481996  0.15938876\n",
      "  -0.3046379   0.39828063  0.80487648 -0.69462498]\n",
      " [-0.00100043 -0.56710161  0.73669033  0.04868083  0.15938876  0.99068038\n",
      "   0.14516898 -0.08241913  0.26028619  0.51602895]\n",
      " [ 0.21929146 -0.28202469 -0.6785701  -0.57794609 -0.3046379   0.14516898\n",
      "  -0.61249417 -0.1417148   0.81758531 -0.17464582]\n",
      " [-0.1587675   0.25638067  0.14691901  0.33673342  0.39828063 -0.08241913\n",
      "  -0.1417148   0.89679732 -0.2961392  -0.0670925 ]\n",
      " [ 0.64460971  0.74370076  0.02618289 -0.06714558  0.80487648  0.26028619\n",
      "   0.81758531 -0.2961392   0.6153814  -0.23107845]\n",
      " [-0.22849717  0.39420788  0.10853407  0.05619415 -0.69462498  0.51602895\n",
      "  -0.17464582 -0.0670925  -0.23107845  0.14642859]]\n"
     ]
    }
   ],
   "source": [
    "N = 10\n",
    "A = np.random.rand(N,N)\n",
    "A_Symm = (A + A.T)/2\n",
    "A_Symm*=2\n",
    "A_Symm-=1\n",
    "print(A_Symm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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0HlP8H+DCiFiWGtgJzKzBuvQw99nA0ZJ+CxzVeo+kIUlLWue8hdbGqZJubR0vbxfYQ0izpurSQtaIeJxqF5tNP18O/EPr9cXAxZON7QRm1mD9vNdXCicwswZzAjOzMgU5Juh7ygnMrMG8nY6ZlcsJzMxK5A0NzaxcEd7Q0MwKVnb+cgIzazIPIc2sTAF4CGlmxSo7fzmBmTWZh5BmVizfhTSzMk2Bsmq1JLDtdn6Go0/9afa4+854KHtMgM9fclz2mFvdOzt7TIDPnrK0lrjve/zEWuIuOfyCWuLWUYDjZZ+6JXtMgOXnHJw95kCG+jbVQtayM5h7YGZN5t0ozKxUpffAvKW0WVOlFrXtQmXuMeduJ2mlpH9Lie0EZtZY1bOQKUeHUipzb/AZ4MepgZ3AzJqsO1WJjqWqyE3r178b7yRJB1MVvb06NbATmFlTtQrbphy0KnOPORZP4kptK3NLmgZ8AfjoZL6CJ/HNmiy9d1V3Ze73A1dGxEpJqW1KS2CSPkRV/iiA24B3R8SzyVcxs/7UP5W5DwVeK+n9wGxguqRnImKi+bL2Q0hJc4HTgKGIOBAYAI5v93Nm1v80Opp0dKhtZe6IeHtE7BER86mGkRe2S16QPgc2CMyQNAjMBB5I/Dkz61dBtZA15ehMSmXu56XtEDIi7pd0LnAv8Cfg6ojY7C5Ba1JvMcB2u83opE1m1gUiurKQNaUy9yafXwBckBI7ZQi5I9Vt0AXAC4FZkjZ7cC4ihiNiKCKGZu64dcq1zazXurOMojYpQ8ijgN9HxKMRsQ64FHhNvc0ys64oPIGl3IW8F3i1pJlUQ8iFwPJaW2Vm9dswB1awlDmwGyQtA24G1gO3AMN1N8zM6pfhDmNPJa0Di4hPAp+suS1m1lX9PTxM4ZX4Zk0VOIGZWcHKHkE6gZk1WekbGjqBmTWZE5iZFSkCRsoeQ9aSwJ5+cDbXn51/rev3Tngye0yAdbPz/y80ulX2kAB8ae/9aon7kh/dX0vcMz89mW2j0q3ZO33LlVR1VA8CePhV+WOu+0mmQO6BmVmxnMDMrEgBuDK3mZUpIDwHZmYlCjyJb2YFK3wOzFWJzJqsC9vppBa2lbSHpKsl3SFphaT57WI7gZk1VmLy6ryXllrY9kLgnIjYDziE8Yt/PIcTmFlTBTA6mnZ0pm1hW0n7A4MRcQ1ARDwTEavbBXYCM2uy7vTA2ha2BfYFnpR0qaRbJJ0jaaBdYE/imzXWpB4lmiNp7E7MwxGxcWPTDIVtB4HXAgdR7QK9FDgJ+NpEjXICM2uqgEhfBzZhZe4MhW1XArdGxN2tn7kMeDVtEpiHkGZNNhppR2faFrYFbgR2kLRT6/3rgBXtAjuBmTVZd+bA2ha2jYgRqorc10m6DRDw1XaBPYQ0a6qIHHcYEy6TVti2dQfypZOJ7QRm1mSFr8R3AjNrrCBGRnrdiI44gZk1lbfTMbOieTsdMytRAOEemJkVKbyhoZkVrPRJfEUNt1ElPQr8X8Kpc4DHsjegPiW1t6S2Qlnt7Ye27hkRO7U/bcskfZ/qu6R4LCIWdXK9OtSSwJIvLi2f6PmqflNSe0tqK5TV3pLaOtX5USIzK5YTmJkVq9cJbLj9KX2lpPaW1FYoq70ltXVK6+kcmJlZJ3rdAzMze96cwMysWD1LYJIWSbpT0l2StlRmqeck7S7ph606dbdLOr3XbUohaaBVHOG7vW7LRCTtIGmZpF+36gEe2us2TUTSh1p/D34l6b8kbdPrNjVZTxJYq9rIl4HXA/sDb2uVVepH64GPRMT+VHt0f6CP2zrW6cAdvW5EgvOB70fEi4GX0cdtljQXOA0YiogDgQHg+N62qtl61QM7BLgrIu6OiLXAt6hqx/WdiHgwIm5uvX6a6h/Y3N62amKS5gFvAJb0ui0TkbQ9cDitwg0RsTYinuxtq9oaBGZIGgRmAg/0uD2N1qsENhe4b8z7lfR5UgBolTo/CLihty1p6zzgY0C/P6m7AHgU+EZruLtE0qxeN2pLIuJ+4Fyqsl8PAn+MiKt726pm8yR+Ikmzge8AH4yIp3rdni2R9EbgkYi4qddtSTAIvAL4SkQcBKxiy2Xne07SjlQjhQXAC4FZkk7sbauarVcJ7H5g9zHv57U+60uStqJKXpdExKW9bk8bhwHHSLqHamj+OkkX97ZJW7QSWBkRG3q0y6gSWr86Cvh9RDwaEeuAS4HX9LhNjdarBHYjsI+kBZKmU02EXt6jtkxIkqjmaO6IiC/2uj3tRMSZETEvIuZT/bn+ICL6spcQEQ8B90l6UeujhSTUAuyhe4FXS5rZ+nuxkD6+6dAEPdkPLCLWSzoFuIrqTs7XI+L2XrQlwWHAO4DbJN3a+uwTEXFlD9s0lZwKXNL6j+xu4N09bs8WRcQNkpYBN1Pdnb4FP1bUU36UyMyK5Ul8MyuWE5iZFcsJzMyK5QRmZsVyAjOzYjmBmVmxnMDMrFj/DwOxw/lzauSLAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(A_Symm)\n",
    "plt.colorbar()\n",
    "plt.savefig(\"Original Matrix\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Eigenvectors and Eigenvalues\n",
    "\n",
    "There are two standard methods to compute the eigenvectors and eigenvalues, one for any general matrices, the second one is specifically for real symmetric matrices\n",
    "Since, this is a symmetric matrix, the eigenvalues  are all always real the set of eigenvectors are orthogonal and hence represent the \"dimensions\" or \"degrees of freedom\" in which the data exists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The eigenvalues are \n",
      "\n",
      "[ 2.17746757  2.02819928  1.5618529  -2.14835398  0.9576927   0.54601796\n",
      " -1.73757565 -0.10570437 -0.88034648 -0.39641073]\n",
      "The eigenvectors are \n",
      "\n",
      "[[-0.40590252 -0.32845758  0.04840936  0.28806855 -0.08100624  0.54761805\n",
      "  -0.00276859 -0.54917049 -0.16996785 -0.07140934]\n",
      " [-0.40134277  0.62980716 -0.09682022 -0.05486476 -0.37679816  0.08432039\n",
      "  -0.49936468 -0.00916899  0.03708314  0.17312288]\n",
      " [ 0.1448123   0.45279265  0.24728691  0.30873667 -0.1055309  -0.08945966\n",
      "   0.45153661 -0.40563666  0.47253523 -0.0817515 ]\n",
      " [ 0.29327398  0.07109931  0.01274624  0.2316118   0.16632393 -0.10547144\n",
      "  -0.55627518 -0.16478143 -0.02845117 -0.69205233]\n",
      " [-0.16921227  0.26857131 -0.02574508  0.56672736  0.38956908 -0.27932628\n",
      "   0.10563638  0.06342577 -0.53947778  0.20344179]\n",
      " [ 0.16620644  0.04174608  0.83356909 -0.14882385  0.22997872  0.14790007\n",
      "  -0.27297487 -0.05236924 -0.08134798  0.30823168]\n",
      " [-0.26138429 -0.24660095  0.09901254  0.49551921  0.1267833   0.06046358\n",
      "  -0.19828155  0.46213957  0.58032281  0.0782974 ]\n",
      " [ 0.16403861  0.35528614 -0.27823386 -0.10408752  0.53109801  0.66223906\n",
      "   0.09112728  0.14084193  0.09758868 -0.01802605]\n",
      " [-0.62152251  0.11240367  0.28014482 -0.33982411  0.25601971 -0.12356776\n",
      "   0.23392453  0.14087289  0.00256769 -0.50172314]\n",
      " [ 0.17149017  0.10878474  0.25645697  0.2232972  -0.49428862  0.34131257\n",
      "   0.22200248  0.49660216 -0.31887032 -0.29108861]]\n"
     ]
    }
   ],
   "source": [
    "W, V = np.linalg.eig(A_Symm)\n",
    "print(\"The eigenvalues are \\n\")\n",
    "print(W)\n",
    "print(\"The eigenvectors are \\n\")\n",
    "print(V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A_Symm  (10, 10)\n",
      "Eigenvectors  (10, 10)\n",
      "Eigenvalues  (10,)\n"
     ]
    }
   ],
   "source": [
    "## Get the dimensions of the matrices\n",
    "\n",
    "print(\"A_Symm \", A_Symm.shape)\n",
    "print(\"Eigenvectors \", V.shape)\n",
    "print(\"Eigenvalues \", W.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The eigenvalues are \n",
      "\n",
      "[-2.14835398 -1.73757565 -0.88034648 -0.39641073 -0.10570437  0.54601796\n",
      "  0.9576927   1.5618529   2.02819928  2.17746757]\n",
      "The eigenvectors are \n",
      "\n",
      "[[ 0.28806855  0.00276859  0.16996785  0.07140934  0.54917049 -0.54761805\n",
      "   0.08100624  0.04840936  0.32845758 -0.40590252]\n",
      " [-0.05486476  0.49936468 -0.03708314 -0.17312288  0.00916899 -0.08432039\n",
      "   0.37679816 -0.09682022 -0.62980716 -0.40134277]\n",
      " [ 0.30873667 -0.45153661 -0.47253523  0.0817515   0.40563666  0.08945966\n",
      "   0.1055309   0.24728691 -0.45279265  0.1448123 ]\n",
      " [ 0.2316118   0.55627518  0.02845117  0.69205233  0.16478143  0.10547144\n",
      "  -0.16632393  0.01274624 -0.07109931  0.29327398]\n",
      " [ 0.56672736 -0.10563638  0.53947778 -0.20344179 -0.06342577  0.27932628\n",
      "  -0.38956908 -0.02574508 -0.26857131 -0.16921227]\n",
      " [-0.14882385  0.27297487  0.08134798 -0.30823168  0.05236924 -0.14790007\n",
      "  -0.22997872  0.83356909 -0.04174608  0.16620644]\n",
      " [ 0.49551921  0.19828155 -0.58032281 -0.0782974  -0.46213957 -0.06046358\n",
      "  -0.1267833   0.09901254  0.24660095 -0.26138429]\n",
      " [-0.10408752 -0.09112728 -0.09758868  0.01802605 -0.14084193 -0.66223906\n",
      "  -0.53109801 -0.27823386 -0.35528614  0.16403861]\n",
      " [-0.33982411 -0.23392453 -0.00256769  0.50172314 -0.14087289  0.12356776\n",
      "  -0.25601971  0.28014482 -0.11240367 -0.62152251]\n",
      " [ 0.2232972  -0.22200248  0.31887032  0.29108861 -0.49660216 -0.34131257\n",
      "   0.49428862  0.25645697 -0.10878474  0.17149017]]\n"
     ]
    }
   ],
   "source": [
    "w, v = np.linalg.eigh(A_Symm)\n",
    "print(\"The eigenvalues are \\n\")\n",
    "print(w)\n",
    "print(\"The eigenvectors are \\n\")\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generating Approximation to the matrices\n",
    "\n",
    "There are many methods to generate approximations to the matrices. We work with the following three methods and configurations\n",
    "1. Eigenvalue Decomposition (2 eigenvectors)\n",
    "2. Eigenvalue Decomposition (4 eigenvectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.51362694  0.02375238 -0.57354865 -0.52947567 -0.56655558 -0.00100043\n",
      "   0.21929146 -0.1587675   0.64460971 -0.22849717]\n",
      " [ 0.02375238  0.85687269  0.86637343 -0.6393855   0.50356679 -0.56710161\n",
      "  -0.28202469  0.25638067  0.74370076  0.39420788]\n",
      " [-0.57354865  0.86637343 -0.20362728  0.41619781 -0.0674289   0.73669033\n",
      "  -0.6785701   0.14691901  0.02618289  0.10853407]\n",
      " [-0.52947567 -0.6393855   0.41619781 -0.61600671 -0.12818749  0.04868083\n",
      "  -0.57794609  0.33673342 -0.06714558  0.05619415]\n",
      " [-0.56655558  0.50356679 -0.0674289  -0.12818749 -0.58481996  0.15938876\n",
      "  -0.3046379   0.39828063  0.80487648 -0.69462498]\n",
      " [-0.00100043 -0.56710161  0.73669033  0.04868083  0.15938876  0.99068038\n",
      "   0.14516898 -0.08241913  0.26028619  0.51602895]\n",
      " [ 0.21929146 -0.28202469 -0.6785701  -0.57794609 -0.3046379   0.14516898\n",
      "  -0.61249417 -0.1417148   0.81758531 -0.17464582]\n",
      " [-0.1587675   0.25638067  0.14691901  0.33673342  0.39828063 -0.08241913\n",
      "  -0.1417148   0.89679732 -0.2961392  -0.0670925 ]\n",
      " [ 0.64460971  0.74370076  0.02618289 -0.06714558  0.80487648  0.26028619\n",
      "   0.81758531 -0.2961392   0.6153814  -0.23107845]\n",
      " [-0.22849717  0.39420788  0.10853407  0.05619415 -0.69462498  0.51602895\n",
      "  -0.17464582 -0.0670925  -0.23107845  0.14642859]]\n"
     ]
    }
   ],
   "source": [
    "## Before moving ahead, first we reconstruct the matrix\n",
    "## The entire matrix through the entrie set of eignevalues and eigenvectors \n",
    "\n",
    "Lambda = np.diag(W)\n",
    "A_recon = V @ Lambda @ np.linalg.inv(V)\n",
    "print(A_recon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(A_recon)\n",
    "plt.colorbar()\n",
    "plt.savefig(\"Reconstructed Matrix\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.57756374 -0.06484069 -0.42963109 -0.30657191 -0.02935964 -0.17471013\n",
      "   0.39530177 -0.38166742  0.4744454  -0.22403974]\n",
      " [-0.06484069  1.15523737  0.45183271 -0.16547488  0.49094257 -0.09192424\n",
      "  -0.08657518  0.31047832  0.68673685 -0.01090828]\n",
      " [-0.42963109  0.45183271  0.46148662  0.15777067  0.19328676  0.09074657\n",
      "  -0.30888766  0.37800368 -0.09275469  0.15397786]\n",
      " [-0.30657191 -0.16547488  0.15777067  0.19753595 -0.0693291   0.11215847\n",
      "  -0.20247933  0.1559877  -0.3806919   0.12519984]\n",
      " [-0.02935964  0.49094257  0.19328676 -0.0693291   0.20864211 -0.0384997\n",
      "  -0.03801937  0.13308938  0.29023069 -0.00392947]\n",
      " [-0.17471013 -0.09192424  0.09074657  0.11215847 -0.0384997   0.06368625\n",
      "  -0.11547691  0.08944893 -0.21541753  0.0712746 ]\n",
      " [ 0.39530177 -0.08657518 -0.30888766 -0.20247933 -0.03801937 -0.11547691\n",
      "   0.2721073  -0.27106198  0.29752379 -0.15201395]\n",
      " [-0.38166742  0.31047832  0.37800368  0.1559877   0.13308938  0.08944893\n",
      "  -0.27106198  0.31460878 -0.14100376  0.13964368]\n",
      " [ 0.4744454   0.68673685 -0.09275469 -0.3806919   0.29023069 -0.21541753\n",
      "   0.29752379 -0.14100376  0.8667599  -0.20728496]\n",
      " [-0.22403974 -0.01090828  0.15397786  0.12519984 -0.00392947  0.0712746\n",
      "  -0.15201395  0.13964368 -0.20728496  0.08803883]]\n"
     ]
    }
   ],
   "source": [
    "## Generating the matrix of the top 2 eigenvalues and the matrices containing the eiegenvectors\n",
    "E_2 = np.zeros((2,2))\n",
    "np.fill_diagonal(E_2,[W[0],W[1]])\n",
    "\n",
    "Q_2 = V[:,0:2]\n",
    "A_2 = np.dot(np.dot(Q_2,E_2),Q_2.T)\n",
    "print(A_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(A_2)\n",
    "plt.colorbar()\n",
    "plt.savefig(\"2-Rank Approximation\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.40294598 -0.03820677 -0.602003   -0.44894653 -0.38203857 -0.01958229\n",
      "   0.09612437 -0.33828723  0.70593472 -0.34284212]\n",
      " [-0.03820677  1.16341157  0.45082871 -0.14010251  0.56163526 -0.23551739\n",
      "  -0.04314143  0.34028386  0.60431894 -0.02336962]\n",
      " [-0.602003    0.45082871  0.35221768  0.00907113 -0.19255312  0.51140368\n",
      "  -0.5993123   0.33958131  0.24084153  0.10492055]\n",
      " [-0.44894653 -0.14010251  0.00907113  0.08254335 -0.35183616  0.20280534\n",
      "  -0.4470707   0.20224099 -0.20602376  0.01919618]\n",
      " [-0.38203857  0.56163526 -0.19255312 -0.35183616 -0.4803308   0.10918011\n",
      "  -0.64531066  0.27100695  0.69271244 -0.28611288]\n",
      " [-0.01958229 -0.23551739  0.51140368  0.20280534  0.10918011  1.1013372\n",
      "   0.17185925 -0.30606668  0.04065575  0.47655307]\n",
      " [ 0.09612437 -0.04314143 -0.5993123  -0.4470707  -0.64531066  0.17185925\n",
      "  -0.24008641 -0.20328246  0.70260621 -0.35006586]\n",
      " [-0.33828723  0.34028386  0.33958131  0.20224099  0.27100695 -0.30606668\n",
      "  -0.20328246  0.41224246 -0.33873399  0.07813066]\n",
      " [ 0.70593472  0.60431894  0.24084153 -0.20602376  0.69271244  0.04065575\n",
      "   0.70260621 -0.33873399  0.74124303  0.06794742]\n",
      " [-0.34284212 -0.02336962  0.10492055  0.01919618 -0.28611288  0.47655307\n",
      "  -0.35006586  0.07813066  0.06794742  0.08364172]]\n"
     ]
    }
   ],
   "source": [
    "## Generating the matrix of the top 2 eigenvalues and the matrices containing the eiegenvectors\n",
    "E_4 = np.zeros((4,4))\n",
    "np.fill_diagonal(E_4,[W[0],W[1],W[2],W[3]])\n",
    "\n",
    "Q_4 = V[ : ,0:4 ]\n",
    "A_4 = np.dot(np.dot(Q_4,E_4),Q_4.T)\n",
    "print(A_4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(A_4)\n",
    "plt.colorbar()\n",
    "plt.savefig(\"4-rank Approximation\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.57295994  0.0138352  -0.6183931  -0.49606276 -0.49527396  0.00548648\n",
      "   0.10341395 -0.18103532  0.65025037 -0.20137206]\n",
      " [ 0.0138352   0.86997323  0.87658254 -0.68764862  0.49985527 -0.54855329\n",
      "  -0.25815399  0.25819296  0.70921591  0.36333997]\n",
      " [-0.6183931   0.87658254  0.01298693  0.43385516 -0.30116133  0.69510655\n",
      "  -0.4595117   0.1820606   0.03747017 -0.03597403]\n",
      " [-0.49606276 -0.68764862  0.43385516 -0.42256837 -0.17159161 -0.03292882\n",
      "  -0.62201084  0.33678114  0.06797758  0.13538735]\n",
      " [-0.49527396  0.49985527 -0.30116133 -0.17159161 -0.3117751   0.22252985\n",
      "  -0.57083631  0.35142362  0.76413926 -0.56333061]\n",
      " [ 0.00548648 -0.54855329  0.69510655 -0.03292882  0.22252985  1.03445767\n",
      "   0.11061813 -0.09239007  0.19801877  0.50054858]\n",
      " [ 0.10341395 -0.25815399 -0.4595117  -0.62201084 -0.57083631  0.11061813\n",
      "  -0.29101008 -0.08553753  0.81020632 -0.32232761]\n",
      " [-0.18103532  0.25819296  0.1820606   0.33678114  0.35142362 -0.09239007\n",
      "  -0.08553753  0.90740695 -0.29023617 -0.08501399]\n",
      " [ 0.65025037  0.70921591  0.03747017  0.06797758  0.76413926  0.19801877\n",
      "   0.81020632 -0.29023617  0.71727185 -0.16651024]\n",
      " [-0.20137206  0.36333997 -0.03597403  0.13538735 -0.56333061  0.50054858\n",
      "  -0.32232761 -0.08501399 -0.16651024  0.29559776]]\n"
     ]
    }
   ],
   "source": [
    "## Generating the matrix of the top 2 eigenvalues and the matrices containing the eiegenvectors\n",
    "E_7 = np.zeros((7,7))\n",
    "np.fill_diagonal(E_7,W[:7])\n",
    "\n",
    "Q_7 = V[ : ,0:7 ]\n",
    "A_7 = np.dot(np.dot(Q_7,E_7),Q_7.T)\n",
    "print(A_7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(A_7)\n",
    "plt.colorbar()\n",
    "plt.savefig(\"7-rank Approximation\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Observations and Comments\n",
    "\n",
    "The above method works only for square matrices, for general rectnagular marices we need to perform Singular Value Decomposition (SVD).\n",
    "To measure the variance of data represented by the approximation, we can measure the frobenius norm of the difference of the approximation matrix and the original matrix.\n",
    "The approximated matrix essentially reduces the complexity in terms of dimensionality of the original matrix, as instead of 10 eigenvectors and values, we using the 'top 2' or 'top 4' vectors according to their 'importance'  "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}