the key things to know for operations on NumPy arrays versus operations on NumPy matrices are:
NumPy matrix is a subclass of NumPy array
NumPy array operations are element-wise (once broadcasting is accounted for)
NumPy matrix operations follow the ordinary rules of linear algebra
some code snippets to illustrate:
>>> from numpy import linalg as LA
>>> import numpy as NP
>>> a1 = NP.matrix("4 3 5; 6 7 8; 1 3 13; 7 21 9")
>>> a1
matrix([[ 4, 3, 5],
[ 6, 7, 8],
[ 1, 3, 13],
[ 7, 21, 9]])
>>> a2 = NP.matrix("7 8 15; 5 3 11; 7 4 9; 6 15 4")
>>> a2
matrix([[ 7, 8, 15],
[ 5, 3, 11],
[ 7, 4, 9],
[ 6, 15, 4]])
>>> a1.shape
(4, 3)
>>> a2.shape
(4, 3)
>>> a2t = a2.T
>>> a2t.shape
(3, 4)
>>> a1 * a2t # same as NP.dot(a1, a2t)
matrix([[127, 84, 85, 89],
[218, 139, 142, 173],
[226, 157, 136, 103],
[352, 197, 214, 393]])
but this operations fails if these two NumPy matrices are converted to arrays:
>>> a1 = NP.array(a1)
>>> a2t = NP.array(a2t)
>>> a1 * a2t
Traceback (most recent call last):
File "<pyshell#277>", line 1, in <module>
a1 * a2t
ValueError: operands could not be broadcast together with shapes (4,3) (3,4)
though using the NP.dot syntax works with arrays; this operations works like matrix multiplication:
>> NP.dot(a1, a2t)
array([[127, 84, 85, 89],
[218, 139, 142, 173],
[226, 157, 136, 103],
[352, 197, 214, 393]])
so do you ever need a NumPy matrix? ie, will a NumPy array suffice for linear algebra computation (provided you know the correct syntax, ie, NP.dot)?
the rule seems to be that if the arguments (arrays) have shapes (m x n) compatible with the a given linear algebra operation, then you are ok, otherwise, NumPy throws.
the only exception i have come across (there are likely others) is calculating matrix inverse.
below are snippets in which i have called a pure linear algebra operation (in fact, from Numpy’s Linear Algebra module) and passed in a NumPy array
determinant of an array:
>>> m = NP.random.randint(0, 10, 16).reshape(4, 4)
>>> m
array([[6, 2, 5, 2],
[8, 5, 1, 6],
[5, 9, 7, 5],
[0, 5, 6, 7]])
>>> type(m)
<type 'numpy.ndarray'>
>>> md = LA.det(m)
>>> md
1772.9999999999995
eigenvectors/eigenvalue pairs:
>>> LA.eig(m)
(array([ 19.703+0.j , 0.097+4.198j, 0.097-4.198j, 5.103+0.j ]),
array([[-0.374+0.j , -0.091+0.278j, -0.091-0.278j, -0.574+0.j ],
[-0.446+0.j , 0.671+0.j , 0.671+0.j , -0.084+0.j ],
[-0.654+0.j , -0.239-0.476j, -0.239+0.476j, -0.181+0.j ],
[-0.484+0.j , -0.387+0.178j, -0.387-0.178j, 0.794+0.j ]]))
matrix norm:
>>>> LA.norm(m)
22.0227
qr factorization:
>>> LA.qr(a1)
(array([[ 0.5, 0.5, 0.5],
[ 0.5, 0.5, -0.5],
[ 0.5, -0.5, 0.5],
[ 0.5, -0.5, -0.5]]),
array([[ 6., 6., 6.],
[ 0., 0., 0.],
[ 0., 0., 0.]]))
matrix rank:
>>> m = NP.random.rand(40).reshape(8, 5)
>>> m
array([[ 0.545, 0.459, 0.601, 0.34 , 0.778],
[ 0.799, 0.047, 0.699, 0.907, 0.381],
[ 0.004, 0.136, 0.819, 0.647, 0.892],
[ 0.062, 0.389, 0.183, 0.289, 0.809],
[ 0.539, 0.213, 0.805, 0.61 , 0.677],
[ 0.269, 0.071, 0.377, 0.25 , 0.692],
[ 0.274, 0.206, 0.655, 0.062, 0.229],
[ 0.397, 0.115, 0.083, 0.19 , 0.701]])
>>> LA.matrix_rank(m)
5
matrix condition:
>>> a1 = NP.random.randint(1, 10, 12).reshape(4, 3)
>>> LA.cond(a1)
5.7093446189400954
inversion requires a NumPy matrix though:
>>> a1 = NP.matrix(a1)
>>> type(a1)
<class 'numpy.matrixlib.defmatrix.matrix'>
>>> a1.I
matrix([[ 0.028, 0.028, 0.028, 0.028],
[ 0.028, 0.028, 0.028, 0.028],
[ 0.028, 0.028, 0.028, 0.028]])
>>> a1 = NP.array(a1)
>>> a1.I
Traceback (most recent call last):
File "<pyshell#230>", line 1, in <module>
a1.I
AttributeError: 'numpy.ndarray' object has no attribute 'I'
but the Moore-Penrose pseudoinverse seems to works just fine
>>> LA.pinv(m)
matrix([[ 0.314, 0.407, -1.008, -0.553, 0.131, 0.373, 0.217, 0.785],
[ 1.393, 0.084, -0.605, 1.777, -0.054, -1.658, 0.069, -1.203],
[-0.042, -0.355, 0.494, -0.729, 0.292, 0.252, 1.079, -0.432],
[-0.18 , 1.068, 0.396, 0.895, -0.003, -0.896, -1.115, -0.666],
[-0.224, -0.479, 0.303, -0.079, -0.066, 0.872, -0.175, 0.901]])
>>> m = NP.array(m)
>>> LA.pinv(m)
array([[ 0.314, 0.407, -1.008, -0.553, 0.131, 0.373, 0.217, 0.785],
[ 1.393, 0.084, -0.605, 1.777, -0.054, -1.658, 0.069, -1.203],
[-0.042, -0.355, 0.494, -0.729, 0.292, 0.252, 1.079, -0.432],
[-0.18 , 1.068, 0.396, 0.895, -0.003, -0.896, -1.115, -0.666],
[-0.224, -0.479, 0.303, -0.079, -0.066, 0.872, -0.175, 0.901]])