numpy_tut.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from __future__ import print_function\n",
    "from pprint import *\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Basic Stuff\n",
    "================"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 3]\n",
      " [4 5 6]]\n"
     ]
    }
   ],
   "source": [
    "a = np.array([[1,2,3],[4,5,6]])\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dtype('int64')"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.dtype"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  2.,  3.],\n",
       "       [ 4.,  5.,  6.]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.astype(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(100000000,)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.shape  ##particularly useful as a no nonsense check "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3, 4, 5, 6])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.reshape((6,))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "total size of new array must be unchanged",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-32-7fca39ab40d6>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0ma\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mreshape\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m10\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;31m##both astype and reshape are not inplace operations\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m: total size of new array must be unchanged"
     ]
    }
   ],
   "source": [
    "a.reshape((10,)) ##both astype and reshape are not inplace operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4],\n",
       "       [5, 6]])"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.reshape((-1,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Aliasing\n",
    "----------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3],\n",
       "       [4, 5, 6]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = a\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1, 10,  3],\n",
       "       [ 4,  5,  6]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b[0][1] = 10\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1, 10,  3],\n",
       "       [ 4,  5,  6]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pass by reference\n",
    "------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 0 ns, sys: 0 ns, total: 0 ns\n",
      "Wall time: 25.3 µs\n"
     ]
    }
   ],
   "source": [
    "def func(a):\n",
    "    a[0,1] = 10\n",
    "a = np.zeros((3,3))\n",
    "func(a)\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Efficeint\n",
    "----------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.arange(100000000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4999999950000000\n",
      "CPU times: user 21.6 s, sys: 0 ns, total: 21.6 s\n",
      "Wall time: 21.6 s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "sum = 0\n",
    "for x in a:\n",
    "    sum+=x\n",
    "print(sum)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 120 ms, sys: 0 ns, total: 120 ms\n",
      "Wall time: 117 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "4999999950000000"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%time\n",
    "np.sum(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Array level operations\n",
    "--------------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 4, 10, 18])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 = np.array([1,2,3])\n",
    "a2 = np.array([4,5,6])\n",
    "a2*a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.,  9.])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = np.linspace(0,9,10) ##unlike most functions in python, the endpoint is included here\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  1.00000000e+00,   2.71828183e+00,   7.38905610e+00,\n",
       "         2.00855369e+01,   5.45981500e+01,   1.48413159e+02,\n",
       "         4.03428793e+02,   1.09663316e+03,   2.98095799e+03,\n",
       "         8.10308393e+03])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.exp(c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/deepak_saini/anaconda3/lib/python3.5/site-packages/ipykernel/__main__.py:2: RuntimeWarning: divide by zero encountered in log\n",
      "  from ipykernel import kernelapp as app\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([[       -inf,  0.        ,  0.69314718,  1.09861229,  1.38629436],\n",
       "       [ 1.60943791,  1.79175947,  1.94591015,  2.07944154,  2.19722458]])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_rs = c.reshape((2,5))\n",
    "np.log(c_rs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Printing np array\n",
    "-------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[   0,    1,    2,    3,    4,    5,    6,    7,    8,    9,   10,\n",
       "          11,   12,   13,   14,   15,   16,   17,   18,   19,   20,   21,\n",
       "          22,   23,   24,   25,   26,   27,   28,   29,   30,   31],\n",
       "       [  32,   33,   34,   35,   36,   37,   38,   39,   40,   41,   42,\n",
       "          43,   44,   45,   46,   47,   48,   49,   50,   51,   52,   53,\n",
       "          54,   55,   56,   57,   58,   59,   60,   61,   62,   63],\n",
       "       [  64,   65,   66,   67,   68,   69,   70,   71,   72,   73,   74,\n",
       "          75,   76,   77,   78,   79,   80,   81,   82,   83,   84,   85,\n",
       "          86,   87,   88,   89,   90,   91,   92,   93,   94,   95],\n",
       "       [  96,   97,   98,   99,  100,  101,  102,  103,  104,  105,  106,\n",
       "         107,  108,  109,  110,  111,  112,  113,  114,  115,  116,  117,\n",
       "         118,  119,  120,  121,  122,  123,  124,  125,  126,  127],\n",
       "       [ 128,  129,  130,  131,  132,  133,  134,  135,  136,  137,  138,\n",
       "         139,  140,  141,  142,  143,  144,  145,  146,  147,  148,  149,\n",
       "         150,  151,  152,  153,  154,  155,  156,  157,  158,  159],\n",
       "       [ 160,  161,  162,  163,  164,  165,  166,  167,  168,  169,  170,\n",
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       "         182,  183,  184,  185,  186,  187,  188,  189,  190,  191],\n",
       "       [ 192,  193,  194,  195,  196,  197,  198,  199,  200,  201,  202,\n",
       "         203,  204,  205,  206,  207,  208,  209,  210,  211,  212,  213,\n",
       "         214,  215,  216,  217,  218,  219,  220,  221,  222,  223],\n",
       "       [ 224,  225,  226,  227,  228,  229,  230,  231,  232,  233,  234,\n",
       "         235,  236,  237,  238,  239,  240,  241,  242,  243,  244,  245,\n",
       "         246,  247,  248,  249,  250,  251,  252,  253,  254,  255],\n",
       "       [ 256,  257,  258,  259,  260,  261,  262,  263,  264,  265,  266,\n",
       "         267,  268,  269,  270,  271,  272,  273,  274,  275,  276,  277,\n",
       "         278,  279,  280,  281,  282,  283,  284,  285,  286,  287],\n",
       "       [ 288,  289,  290,  291,  292,  293,  294,  295,  296,  297,  298,\n",
       "         299,  300,  301,  302,  303,  304,  305,  306,  307,  308,  309,\n",
       "         310,  311,  312,  313,  314,  315,  316,  317,  318,  319],\n",
       "       [ 320,  321,  322,  323,  324,  325,  326,  327,  328,  329,  330,\n",
       "         331,  332,  333,  334,  335,  336,  337,  338,  339,  340,  341,\n",
       "         342,  343,  344,  345,  346,  347,  348,  349,  350,  351],\n",
       "       [ 352,  353,  354,  355,  356,  357,  358,  359,  360,  361,  362,\n",
       "         363,  364,  365,  366,  367,  368,  369,  370,  371,  372,  373,\n",
       "         374,  375,  376,  377,  378,  379,  380,  381,  382,  383],\n",
       "       [ 384,  385,  386,  387,  388,  389,  390,  391,  392,  393,  394,\n",
       "         395,  396,  397,  398,  399,  400,  401,  402,  403,  404,  405,\n",
       "         406,  407,  408,  409,  410,  411,  412,  413,  414,  415],\n",
       "       [ 416,  417,  418,  419,  420,  421,  422,  423,  424,  425,  426,\n",
       "         427,  428,  429,  430,  431,  432,  433,  434,  435,  436,  437,\n",
       "         438,  439,  440,  441,  442,  443,  444,  445,  446,  447],\n",
       "       [ 448,  449,  450,  451,  452,  453,  454,  455,  456,  457,  458,\n",
       "         459,  460,  461,  462,  463,  464,  465,  466,  467,  468,  469,\n",
       "         470,  471,  472,  473,  474,  475,  476,  477,  478,  479],\n",
       "       [ 480,  481,  482,  483,  484,  485,  486,  487,  488,  489,  490,\n",
       "         491,  492,  493,  494,  495,  496,  497,  498,  499,  500,  501,\n",
       "         502,  503,  504,  505,  506,  507,  508,  509,  510,  511],\n",
       "       [ 512,  513,  514,  515,  516,  517,  518,  519,  520,  521,  522,\n",
       "         523,  524,  525,  526,  527,  528,  529,  530,  531,  532,  533,\n",
       "         534,  535,  536,  537,  538,  539,  540,  541,  542,  543],\n",
       "       [ 544,  545,  546,  547,  548,  549,  550,  551,  552,  553,  554,\n",
       "         555,  556,  557,  558,  559,  560,  561,  562,  563,  564,  565,\n",
       "         566,  567,  568,  569,  570,  571,  572,  573,  574,  575],\n",
       "       [ 576,  577,  578,  579,  580,  581,  582,  583,  584,  585,  586,\n",
       "         587,  588,  589,  590,  591,  592,  593,  594,  595,  596,  597,\n",
       "         598,  599,  600,  601,  602,  603,  604,  605,  606,  607],\n",
       "       [ 608,  609,  610,  611,  612,  613,  614,  615,  616,  617,  618,\n",
       "         619,  620,  621,  622,  623,  624,  625,  626,  627,  628,  629,\n",
       "         630,  631,  632,  633,  634,  635,  636,  637,  638,  639],\n",
       "       [ 640,  641,  642,  643,  644,  645,  646,  647,  648,  649,  650,\n",
       "         651,  652,  653,  654,  655,  656,  657,  658,  659,  660,  661,\n",
       "         662,  663,  664,  665,  666,  667,  668,  669,  670,  671],\n",
       "       [ 672,  673,  674,  675,  676,  677,  678,  679,  680,  681,  682,\n",
       "         683,  684,  685,  686,  687,  688,  689,  690,  691,  692,  693,\n",
       "         694,  695,  696,  697,  698,  699,  700,  701,  702,  703],\n",
       "       [ 704,  705,  706,  707,  708,  709,  710,  711,  712,  713,  714,\n",
       "         715,  716,  717,  718,  719,  720,  721,  722,  723,  724,  725,\n",
       "         726,  727,  728,  729,  730,  731,  732,  733,  734,  735],\n",
       "       [ 736,  737,  738,  739,  740,  741,  742,  743,  744,  745,  746,\n",
       "         747,  748,  749,  750,  751,  752,  753,  754,  755,  756,  757,\n",
       "         758,  759,  760,  761,  762,  763,  764,  765,  766,  767],\n",
       "       [ 768,  769,  770,  771,  772,  773,  774,  775,  776,  777,  778,\n",
       "         779,  780,  781,  782,  783,  784,  785,  786,  787,  788,  789,\n",
       "         790,  791,  792,  793,  794,  795,  796,  797,  798,  799],\n",
       "       [ 800,  801,  802,  803,  804,  805,  806,  807,  808,  809,  810,\n",
       "         811,  812,  813,  814,  815,  816,  817,  818,  819,  820,  821,\n",
       "         822,  823,  824,  825,  826,  827,  828,  829,  830,  831],\n",
       "       [ 832,  833,  834,  835,  836,  837,  838,  839,  840,  841,  842,\n",
       "         843,  844,  845,  846,  847,  848,  849,  850,  851,  852,  853,\n",
       "         854,  855,  856,  857,  858,  859,  860,  861,  862,  863],\n",
       "       [ 864,  865,  866,  867,  868,  869,  870,  871,  872,  873,  874,\n",
       "         875,  876,  877,  878,  879,  880,  881,  882,  883,  884,  885,\n",
       "         886,  887,  888,  889,  890,  891,  892,  893,  894,  895],\n",
       "       [ 896,  897,  898,  899,  900,  901,  902,  903,  904,  905,  906,\n",
       "         907,  908,  909,  910,  911,  912,  913,  914,  915,  916,  917,\n",
       "         918,  919,  920,  921,  922,  923,  924,  925,  926,  927],\n",
       "       [ 928,  929,  930,  931,  932,  933,  934,  935,  936,  937,  938,\n",
       "         939,  940,  941,  942,  943,  944,  945,  946,  947,  948,  949,\n",
       "         950,  951,  952,  953,  954,  955,  956,  957,  958,  959],\n",
       "       [ 960,  961,  962,  963,  964,  965,  966,  967,  968,  969,  970,\n",
       "         971,  972,  973,  974,  975,  976,  977,  978,  979,  980,  981,\n",
       "         982,  983,  984,  985,  986,  987,  988,  989,  990,  991],\n",
       "       [ 992,  993,  994,  995,  996,  997,  998,  999, 1000, 1001, 1002,\n",
       "        1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013,\n",
       "        1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.arange(1024).reshape((32,32))\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "np.set_printoptions(threshold=np.nan)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Axes\n",
    "-------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0,  1,  2,  3,  4],\n",
       "       [ 5,  6,  7,  8,  9],\n",
       "       [10, 11, 12, 13, 14],\n",
       "       [15, 16, 17, 18, 19]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.arange(20).reshape((4,5))\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([30, 34, 38, 42, 46])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(a,axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  2.,   7.,  12.,  17.])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(a,axis=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Upcasting\n",
    "------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  2.  3.  4.]\n",
      "[1 2 3 4]\n"
     ]
    }
   ],
   "source": [
    "a = np.array([1,2,3,4],dtype=np.float64)\n",
    "b = np.array([1,2,3,4])\n",
    "print(a)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dtype('float64')"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = a+b\n",
    "c.dtype"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  3.,   6.,   9.,  12.])"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a+=b ##Inplace operation\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "Cannot cast ufunc add output from dtype('float64') to dtype('int64') with casting rule 'same_kind'",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-59-7ff1e479bdab>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mb\u001b[0m\u001b[1;33m+=\u001b[0m\u001b[0ma\u001b[0m \u001b[1;31m##Data types is fundamental property of an array, you can't modify that\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m: Cannot cast ufunc add output from dtype('float64') to dtype('int64') with casting rule 'same_kind'"
     ]
    }
   ],
   "source": [
    "b+=a ##Data types is fundamental property of an array, you can't modify that"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Slicing\n",
    "---------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0.,   3.,   6.,   9.,  12.,  15.,  18.,  21.,  24.,  27.],\n",
       "       [  2.,   5.,   8.,  11.,  14.,  17.,  20.,  23.,  26.,  29.],\n",
       "       [  4.,   7.,  10.,  13.,  16.,  19.,  22.,  25.,  28.,  31.],\n",
       "       [  6.,   9.,  12.,  15.,  18.,  21.,  24.,  27.,  30.,  33.],\n",
       "       [  8.,  11.,  14.,  17.,  20.,  23.,  26.,  29.,  32.,  35.]])"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(i,j):\n",
    "    return 2*i+3*j\n",
    "a = np.fromfunction(f,(5,10))\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "12\n",
      "12\n"
     ]
    }
   ],
   "source": [
    "print(a[1,2])\n",
    "print(a[1][2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[0,1:4] ##remember the end point is not incldedd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[12, 13, 14, 15],\n",
       "       [22, 23, 24, 25]])"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[1:3,2:6]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0.,   6.,  12.,  18.,  24.],\n",
       "       [  2.,   8.,  14.,  20.,  26.],\n",
       "       [  4.,  10.,  16.,  22.,  28.],\n",
       "       [  6.,  12.,  18.,  24.,  30.],\n",
       "       [  8.,  14.,  20.,  26.,  32.]])"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[:,0:10:2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "33.0"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[-2,-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  8.,  11.,  14.,  17.,  20.,  23.,  26.,  29.,  32.,  35.],\n",
       "       [  6.,   9.,  12.,  15.,  18.,  21.,  24.,  27.,  30.,  33.],\n",
       "       [  4.,   7.,  10.,  13.,  16.,  19.,  22.,  25.,  28.,  31.],\n",
       "       [  2.,   5.,   8.,  11.,  14.,  17.,  20.,  23.,  26.,  29.],\n",
       "       [  0.,   3.,   6.,   9.,  12.,  15.,  18.,  21.,  24.,  27.]])"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[: :-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.0,\n",
       " 3.0,\n",
       " 6.0,\n",
       " 9.0,\n",
       " 12.0,\n",
       " 15.0,\n",
       " 18.0,\n",
       " 21.0,\n",
       " 24.0,\n",
       " 27.0,\n",
       " 2.0,\n",
       " 5.0,\n",
       " 8.0,\n",
       " 11.0,\n",
       " 14.0,\n",
       " 17.0,\n",
       " 20.0,\n",
       " 23.0,\n",
       " 26.0,\n",
       " 29.0,\n",
       " 4.0,\n",
       " 7.0,\n",
       " 10.0,\n",
       " 13.0,\n",
       " 16.0,\n",
       " 19.0,\n",
       " 22.0,\n",
       " 25.0,\n",
       " 28.0,\n",
       " 31.0,\n",
       " 6.0,\n",
       " 9.0,\n",
       " 12.0,\n",
       " 15.0,\n",
       " 18.0,\n",
       " 21.0,\n",
       " 24.0,\n",
       " 27.0,\n",
       " 30.0,\n",
       " 33.0,\n",
       " 8.0,\n",
       " 11.0,\n",
       " 14.0,\n",
       " 17.0,\n",
       " 20.0,\n",
       " 23.0,\n",
       " 26.0,\n",
       " 29.0,\n",
       " 32.0,\n",
       " 35.0]"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(a.flat) ##returns an iterator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0.,   3.,   6.,   9.,  12.,  15.,  18.,  21.,  24.,  27.],\n",
       "       [  2.,   5.,   8.,  11.,  14.,  17.,  20.,  23.,  26.,  29.],\n",
       "       [  4.,   7.,  10.,  13.,  16.,  19.,  22.,  25.,  28.,  31.],\n",
       "       [  6.,   9.,  12.,  15.,  18.,  21.,  24.,  27.,  30.,  33.],\n",
       "       [  8.,  11.,  14.,  17.,  20.,  23.,  26.,  29.,  32.,  35.]])"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[[  0.,   3.,   6.,   9.,  12.,  15.,  18.,  21.,  24.,  27.],\n",
       "        [  2.,   5.,   8.,  11.,  14.,  17.,  20.,  23.,  26.,  29.]],\n",
       "\n",
       "       [[  4.,   7.,  10.,  13.,  16.,  19.,  22.,  25.,  28.,  31.],\n",
       "        [  6.,   9.,  12.,  15.,  18.,  21.,  24.,  27.,  30.,  33.]]])"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i = np.array([[0,1],[2,3]])\n",
    "a[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  2.],\n",
       "       [ 7.,  9.]])"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j = np.array([[0,0],[1,1]])\n",
    "a[i,j]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.,  2.],\n",
       "       [ 5.,  5.]])"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[1,j]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0.,   3.,   6.,   9.,  12.,  15.,  18.,  21.,  24.,  27.],\n",
       "       [  0.,   5.,   8.,  11.,  14.,  17.,  20.,  23.,  26.,  29.],\n",
       "       [  4.,   0.,  10.,  13.,  16.,  19.,  22.,  25.,  28.,  31.],\n",
       "       [  6.,   0.,  12.,  15.,  18.,  21.,  24.,  27.,  30.,  33.],\n",
       "       [  8.,  11.,  14.,  17.,  20.,  23.,  26.,  29.,  32.,  35.]])"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = np.copy(a)\n",
    "b[i,j] = 0\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 1, 2, 3, 4],\n",
       "       [0, 1, 2, 3, 4],\n",
       "       [0, 1, 2, 3, 4],\n",
       "       [0, 1, 2, 3, 4],\n",
       "       [0, 1, 2, 3, 4],\n",
       "       [0, 1, 2, 3, 4],\n",
       "       [0, 1, 2, 3, 4],\n",
       "       [0, 1, 2, 3, 4],\n",
       "       [0, 1, 2, 3, 4],\n",
       "       [0, 1, 2, 3, 4]])"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#fun way to transpose an array\n",
    "x = np.array([[i for i in range(5)]])\n",
    "i = np.repeat(x,10,axis=0)\n",
    "i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 0, 0, 0, 0],\n",
       "       [1, 1, 1, 1, 1],\n",
       "       [2, 2, 2, 2, 2],\n",
       "       [3, 3, 3, 3, 3],\n",
       "       [4, 4, 4, 4, 4],\n",
       "       [5, 5, 5, 5, 5],\n",
       "       [6, 6, 6, 6, 6],\n",
       "       [7, 7, 7, 7, 7],\n",
       "       [8, 8, 8, 8, 8],\n",
       "       [9, 9, 9, 9, 9]])"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = np.arange(10).reshape((10,-1))\n",
    "j = np.repeat(y,5,axis=1)\n",
    "j"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0.,   2.,   4.,   6.,   8.],\n",
       "       [  3.,   5.,   7.,   9.,  11.],\n",
       "       [  6.,   8.,  10.,  12.,  14.],\n",
       "       [  9.,  11.,  13.,  15.,  17.],\n",
       "       [ 12.,  14.,  16.,  18.,  20.],\n",
       "       [ 15.,  17.,  19.,  21.,  23.],\n",
       "       [ 18.,  20.,  22.,  24.,  26.],\n",
       "       [ 21.,  23.,  25.,  27.,  29.],\n",
       "       [ 24.,  26.,  28.,  30.,  32.],\n",
       "       [ 27.,  29.,  31.,  33.,  35.]])"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[i,j]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  0.   3.   6.   9.  12.  15.  18.  21.  24.  27.]\n",
      " [  2.   5.   8.  11.  14.  17.  20.  23.  26.  29.]\n",
      " [  4.   7.  10.  13.  16.  19.  22.  25.  28.  31.]\n",
      " [  6.   9.  12.  15.  18.  21.  24.  27.  30.  33.]\n",
      " [  8.  11.  14.  17.  20.  23.  26.  29.  32.  35.]]\n"
     ]
    }
   ],
   "source": [
    "##Boolean indexing\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[False, False, False, False,  True,  True,  True,  True,  True,\n",
       "         True],\n",
       "       [False, False, False,  True,  True,  True,  True,  True,  True,\n",
       "         True],\n",
       "       [False, False, False,  True,  True,  True,  True,  True,  True,\n",
       "         True],\n",
       "       [False, False,  True,  True,  True,  True,  True,  True,  True,\n",
       "         True],\n",
       "       [False,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "         True]], dtype=bool)"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = a>10\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 12.,  15.,  18.,  21.,  24.,  27.,  11.,  14.,  17.,  20.,  23.,\n",
       "        26.,  29.,  13.,  16.,  19.,  22.,  25.,  28.,  31.,  12.,  15.,\n",
       "        18.,  21.,  24.,  27.,  30.,  33.,  11.,  14.,  17.,  20.,  23.,\n",
       "        26.,  29.,  32.,  35.])"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[b]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  0.,   3.,   6.,   9.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.],\n",
       "       [  2.,   5.,   8.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.],\n",
       "       [  4.,   7.,  10.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.],\n",
       "       [  6.,   9.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.],\n",
       "       [  8.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.,  -1.]])"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = a.copy()\n",
    "c[b] = -1\n",
    "c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Stacking\n",
    "-----------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 2 4 6 8]\n",
      "[0 1 2 3 4]\n"
     ]
    }
   ],
   "source": [
    "x = np.arange(0,10,2)                     \n",
    "y = np.arange(5)                          \n",
    "print(x)\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0, 2, 4, 6, 8],\n",
       "       [0, 1, 2, 3, 4]])"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.vstack([x,y])\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 2, 4, 6, 8, 0, 1, 2, 3, 4])"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = np.hstack([x,y]) \n",
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "More Relevant Stuff\n",
    "====================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "np arrays as matrices\n",
    "-----------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a = np.array([[1,2],[3,4]])\n",
    "b = np.array([[4,5],[6,7]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 3],\n",
       "       [2, 4]])"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[16, 19],\n",
       "       [36, 43]])"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(a,b) #matrix mult"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Linear algebra ops : numpy linalg\n",
    "-----------------------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0,  1,  2,  3,  4],\n",
       "       [ 5,  6,  7,  8,  9],\n",
       "       [10, 11, 12, 13, 14],\n",
       "       [15, 16, 17, 18, 19],\n",
       "       [20, 21, 22, 23, 24]])"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.arange(25).reshape((5,5))\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "i5 = np.eye(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.all(a.dot(i5)==a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "60"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.trace(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "ename": "LinAlgError",
     "evalue": "Singular matrix",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mLinAlgError\u001b[0m                               Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-125-5c2063aa5f16>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mlinalg\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0minv\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0ma\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;32m/home/deepak_saini/anaconda3/lib/python3.5/site-packages/numpy/linalg/linalg.py\u001b[0m in \u001b[0;36minv\u001b[1;34m(a)\u001b[0m\n\u001b[0;32m    524\u001b[0m     \u001b[0msignature\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;34m'D->D'\u001b[0m \u001b[1;32mif\u001b[0m \u001b[0misComplexType\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mt\u001b[0m\u001b[1;33m)\u001b[0m \u001b[1;32melse\u001b[0m \u001b[1;34m'd->d'\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    525\u001b[0m     \u001b[0mextobj\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mget_linalg_error_extobj\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0m_raise_linalgerror_singular\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 526\u001b[1;33m     \u001b[0mainv\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0m_umath_linalg\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0minv\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0ma\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0msignature\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0msignature\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mextobj\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mextobj\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    527\u001b[0m     \u001b[1;32mreturn\u001b[0m \u001b[0mwrap\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mainv\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mastype\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mresult_t\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcopy\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;32mFalse\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    528\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m/home/deepak_saini/anaconda3/lib/python3.5/site-packages/numpy/linalg/linalg.py\u001b[0m in \u001b[0;36m_raise_linalgerror_singular\u001b[1;34m(err, flag)\u001b[0m\n\u001b[0;32m     88\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     89\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0m_raise_linalgerror_singular\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0merr\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mflag\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 90\u001b[1;33m     \u001b[1;32mraise\u001b[0m \u001b[0mLinAlgError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"Singular matrix\"\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     91\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     92\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0m_raise_linalgerror_nonposdef\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0merr\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mflag\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mLinAlgError\u001b[0m: Singular matrix"
     ]
    }
   ],
   "source": [
    "np.linalg.inv(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.matrix_rank(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([  6.39116499e+01,  -3.91164992e+00,  -4.06523933e-15,\n",
       "          2.58443073e-16,   1.51528038e-15]),\n",
       " array([[-0.0851802 ,  0.67779864, -0.46549901,  0.18772213, -0.00207481],\n",
       "        [-0.23825372,  0.36348873,  0.55597991, -0.0514454 , -0.41636356],\n",
       "        [-0.39132723,  0.04917881, -0.15590518, -0.65194893,  0.2955311 ],\n",
       "        [-0.54440074, -0.2651311 ,  0.50586667,  0.70734553,  0.66632773],\n",
       "        [-0.69747425, -0.57944101, -0.44044239, -0.19167334, -0.54342046]]))"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.eig(a)"
   ]
  }
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