import numpy as np
#np.random.rand generates random values between 0 and 1 of size (n,m)
vec_1 = np.random.rand(1, 4) #write commas as in writing, with a space after the comma
vec_2 = np.random.rand(4, 4) #use underscores and descriptive names
print 'Vec_1 is equal to \n', vec_1
print 'Vec_2 is equal to \n', vec_2
Vec_1 is equal to [[ 0.99899858 0.73141335 0.52014911 0.26303083]] Vec_2 is equal to [[ 0.72321931 0.28241295 0.69711204 0.71115872] [ 0.75777035 0.67067145 0.81723818 0.74861268] [ 0.98773623 0.30544845 0.64481952 0.69253541] [ 0.68356857 0.59590889 0.07894407 0.7613203 ]]
Both vectors are also matrices.
print np.trace(vec_2) #Trace
2.80003058694
print vec_2.T #Transpose
[[ 0.72321931 0.75777035 0.98773623 0.68356857] [ 0.28241295 0.67067145 0.30544845 0.59590889] [ 0.69711204 0.81723818 0.64481952 0.07894407] [ 0.71115872 0.74861268 0.69253541 0.7613203 ]]
print np.diag(vec_2)
[ 0.72321931 0.67067145 0.64481952 0.7613203 ]
print np.dot(vec_2, vec_2)
[[ 1.91173889 1.0303704 1.24061675 1.74993598] [ 2.37519295 1.35953345 1.66241882 2.17686856] [ 2.05611674 1.09345219 1.40865074 1.90490144] [ 1.54432262 1.07049858 1.07452978 1.56651087]]
print np.dot(vec_1, vec_1.T)
[[ 1.87270397]]
'''
This is another way to make block comments in python,
Bug warning!!!!
'''
vec_1 = np.random.rand(1, 4) #write commas as in writing, with a space after the comma
vec_1_ = np.random.rand(4)
print vec_1 #this is a 2D array (matrix) of size 1x4
print vec_1_ #this is a 1D array of size 4
[[ 0.34399734 0.37411323 0.50543426 0.6863573 ]] [ 0.95862379 0.06110035 0.92039397 0.2059746 ]
#This can generate confusion and some bugs
print np.dot(vec_1_, vec_1_) #This works! And returns a scalar
1.81224340453
print np.dot(vec_1, vec_1) #This doesn't work
--------------------------------------------------------------------------- ValueError Traceback (most recent call last) <ipython-input-121-11aadd02bde4> in <module>() ----> 1 print np.dot(vec_1, vec_1) #This doesn't work ValueError: shapes (1,4) and (1,4) not aligned: 4 (dim 1) != 1 (dim 0)
# When np.dot detects a 1D array, operates as vector's inner product.
# When it detects 2D arrays it operates as matrix multiplication
# Is not possible to multiply a 1x4 matrix by a 1x4 matrix
# We can use the transpose
print np.dot(vec_1, vec_1.T) #This works, but the output is not a scalar is a 1x1 Matrix (yes, it sounds ridiculous)
[[ 0.98484501]]
Let's play with indexes.
print vec_1_
print vec_1_[::-1] #inverts the order
[ 0.95862379 0.06110035 0.92039397 0.2059746 ] [ 0.2059746 0.92039397 0.06110035 0.95862379]
print vec_1_[0] #first element
print vec_1_[-1] #last element
0.958623785967 0.20597460166
print vec_1_>0.5
[ True False True False]
print np.nonzero(vec_1_>0.5)
(array([0, 2], dtype=int64),)
print np.nonzero(vec_1_>0.5)
(array([0, 2], dtype=int64),)
print vec_1_[np.nonzero(vec_1_>0.5)]
[ 0.95862379 0.92039397]
print np.hstack((vec_1_, vec_1_))
[ 0.95862379 0.06110035 0.92039397 0.2059746 0.95862379 0.06110035 0.92039397 0.2059746 ]
print np.vstack((vec_1_, vec_1_))
[[ 0.95862379 0.06110035 0.92039397 0.2059746 ] [ 0.95862379 0.06110035 0.92039397 0.2059746 ]]
print np.vstack((vec_1_, vec_1_)).shape
(2L, 4L)
print vec_2
[[ 0.72321931 0.28241295 0.69711204 0.71115872] [ 0.75777035 0.67067145 0.81723818 0.74861268] [ 0.98773623 0.30544845 0.64481952 0.69253541] [ 0.68356857 0.59590889 0.07894407 0.7613203 ]]
print vec_2+vec_2 #elementwise sum
[[ 1.44643862 0.56482589 1.39422409 1.42231744] [ 1.5155407 1.34134291 1.63447635 1.49722537] [ 1.97547245 0.61089691 1.28963905 1.38507082] [ 1.36713713 1.19181778 0.15788814 1.5226406 ]]
print vec_2*vec_2 #elementwise product (not Matrix Multiplication!)
[[ 0.52304617 0.07975707 0.4859652 0.50574672] [ 0.57421591 0.4498002 0.66787824 0.56042095] [ 0.97562285 0.09329876 0.41579222 0.47960529] [ 0.46726599 0.35510741 0.00623217 0.5796086 ]]
print np.dot(vec_2, vec_2)
[[ 1.91173889 1.0303704 1.24061675 1.74993598] [ 2.37519295 1.35953345 1.66241882 2.17686856] [ 2.05611674 1.09345219 1.40865074 1.90490144] [ 1.54432262 1.07049858 1.07452978 1.56651087]]
print np.linalg.det(vec_2)
-0.0561744297093
print np.power(vec_2, 2)
[[ 0.52304617 0.07975707 0.4859652 0.50574672] [ 0.57421591 0.4498002 0.66787824 0.56042095] [ 0.97562285 0.09329876 0.41579222 0.47960529] [ 0.46726599 0.35510741 0.00623217 0.5796086 ]]
print np.linalg.matrix_power(vec_2, 2)
[[ 1.91173889 1.0303704 1.24061675 1.74993598] [ 2.37519295 1.35953345 1.66241882 2.17686856] [ 2.05611674 1.09345219 1.40865074 1.90490144] [ 1.54432262 1.07049858 1.07452978 1.56651087]]
print np.linalg.matrix_power(vec_2, 5)
[[ 28.14146233 16.36148772 19.12752395 26.49869503] [ 36.04754598 20.95189104 24.50016832 33.94052391] [ 30.64513248 17.8163811 20.82779782 28.85604704] [ 25.11603742 14.60142921 17.07471497 23.6465306 ]]
import pylab as plt
%matplotlib inline
circle_points = np.random.rand(1000,2)*2 -1 #this shifts the values to generate values between -1 and 1
print circle_points[1:5, :]
[[-0.23245839 0.34721948] [ 0.9310904 0.22913624] [ 0.58974444 0.01265035] [-0.81659372 -0.93399382]]
plt.scatter(circle_points[:,0], circle_points[:,1])
<matplotlib.collections.PathCollection at 0xb6ecd68>
dist_points = np.power(circle_points[:,0], 2) + np.power(circle_points[:,1], 2)
dist_points = np.sqrt(dist_points)
print dist_points [1:5]
[ 0.41784958 0.95887056 0.58988011 1.2406328 ]
print np.nonzero(dist_points<1)
(array([ 0, 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 53, 54, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 68, 70, 71, 72, 73, 74, 76, 77, 78, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 94, 95, 96, 99, 100, 102, 103, 104, 105, 108, 109, 110, 112, 113, 114, 116, 117, 118, 120, 122, 123, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 137, 138, 140, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 157, 159, 160, 161, 162, 163, 164, 165, 166, 169, 171, 172, 173, 175, 176, 177, 178, 179, 181, 182, 183, 186, 187, 190, 191, 192, 193, 194, 196, 198, 199, 201, 203, 204, 205, 206, 207, 208, 209, 210, 212, 213, 214, 215, 217, 218, 220, 221, 223, 227, 228, 229, 230, 231, 232, 233, 235, 237, 240, 242, 243, 244, 245, 246, 248, 249, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 264, 265, 267, 269, 270, 272, 275, 276, 277, 280, 282, 283, 285, 286, 287, 289, 290, 293, 294, 295, 296, 298, 301, 302, 304, 306, 307, 309, 310, 311, 314, 315, 316, 317, 318, 320, 321, 322, 324, 325, 326, 327, 328, 329, 330, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 347, 348, 350, 351, 352, 353, 354, 355, 356, 358, 359, 360, 361, 365, 366, 367, 368, 369, 370, 371, 372, 373, 375, 377, 378, 379, 380, 381, 382, 385, 386, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 399, 400, 401, 402, 404, 406, 407, 410, 411, 413, 414, 416, 418, 419, 420, 421, 423, 424, 425, 426, 428, 429, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 447, 450, 451, 454, 455, 456, 457, 458, 459, 460, 461, 463, 464, 465, 466, 467, 468, 470, 471, 472, 473, 476, 479, 480, 481, 483, 485, 486, 487, 488, 489, 490, 492, 493, 495, 497, 498, 499, 500, 501, 502, 503, 505, 506, 507, 508, 509, 511, 512, 513, 514, 515, 516, 517, 518, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 534, 536, 537, 539, 540, 542, 543, 544, 545, 546, 547, 548, 549, 551, 552, 553, 556, 558, 560, 561, 562, 563, 564, 565, 566, 567, 569, 570, 571, 574, 575, 577, 578, 579, 580, 581, 582, 583, 587, 588, 590, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 624, 625, 626, 629, 630, 631, 633, 634, 637, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 651, 652, 654, 656, 657, 660, 662, 663, 666, 667, 668, 669, 670, 671, 672, 676, 677, 679, 683, 684, 688, 689, 690, 691, 692, 693, 695, 696, 698, 699, 700, 701, 704, 705, 707, 708, 710, 711, 712, 713, 714, 715, 716, 718, 719, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 737, 739, 740, 742, 743, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 759, 760, 761, 762, 763, 764, 765, 766, 768, 773, 774, 776, 777, 778, 779, 780, 781, 782, 783, 785, 786, 787, 788, 789, 791, 792, 793, 795, 796, 797, 798, 799, 801, 802, 805, 806, 808, 809, 810, 813, 814, 815, 816, 817, 818, 819, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 832, 833, 834, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 847, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 865, 866, 867, 868, 869, 870, 871, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 888, 890, 891, 892, 893, 895, 896, 898, 899, 900, 901, 902, 903, 904, 906, 907, 910, 913, 919, 920, 921, 922, 923, 924, 925, 927, 928, 929, 930, 932, 933, 934, 936, 937, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 957, 958, 959, 960, 962, 963, 964, 966, 967, 968, 969, 970, 971, 974, 975, 976, 977, 978, 979, 980, 981, 983, 984, 985, 986, 989, 990, 991, 992, 993, 994, 995, 997, 999], dtype=int64),)
inner_circle = circle_points[np.nonzero(dist_points<1)[0], :] #ask me about the zero
plt.scatter(inner_circle[:,0], inner_circle[:,1])
plt.gca().set_aspect('equal', adjustable='box') #without this it doesn't look like a circle
print inner_circle.shape
(758L, 2L)
transf_matrix = np.array([[2,1], [1,2]])
print transf_matrix
[[2 1] [1 2]]
transf_dots = np.dot(transf_matrix, inner_circle.T).T
print transf_dots.shape
(758L, 2L)
plt.scatter(transf_dots[:,0], transf_dots[:,1])
plt.gca().set_aspect('equal', adjustable='box') #without this it doesn't look like a circle
print np.linalg.eig(transf_matrix)
(array([ 3., 1.]), array([[ 0.70710678, -0.70710678], [ 0.70710678, 0.70710678]]))