np.linalg

@TOC

np.linalg.inv

np.linalg.inv(a)

计算一个矩阵的逆

Compute the (multiplicative) inverse of a matrix.

Given a square matrix a, return the matrix ainv satisfying dot(a, ainv) = dot(ainv, a) = eye(a.shape[0]).

import numpy as np
from numpy.linalg import inv
a = np.array([[1., 2.], [3., 4.]])
ainv = inv(a)
np.allclose(np.dot(a, ainv), np.eye(2))
np.allclose(np.dot(ainv, a), np.eye(2))
'''
[[-2.   1. ]
 [ 1.5 -0.5]]
True
True
'''


# 可以同时求多个方阵的逆
x1 = np.array([[1., 2.], [3., 4.]])
x2 = np.array([[1, 3], [3, 5]])
a = np.array([[[1., 2.], [3., 4.]], [[1, 3], [3, 5]]])
print(inv(a))
print(inv(x1))
print(inv(x2))
'''
[[[-2.    1.  ]
  [ 1.5  -0.5 ]]

 [[-1.25  0.75]
  [ 0.75 -0.25]]]


[[-2.   1. ]
 [ 1.5 -0.5]]
[[-1.25  0.75]
 [ 0.75 -0.25]]
'''

np.linalg.norm

np.linalg.norm

计算矩阵或者向量的逆

Matrix or vector norm.

This function is able to return one of eight different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ord parameter.

import numpy as np
import numpy.linalg as LA

a = np.arange(9)
print(a)
print(LA.norm(a, np.inf))  # 无穷范数
print(LA.norm(a, -np.inf))
print(LA.norm(a, 1))  # 1范数
print(LA.norm(a, 2))
'''
[0 1 2 3 4 5 6 7 8]
8.0
0.0
36.0
14.2828568570857
'''

ord	norm for matrices	norm for vectors
None	Frobenius norm	2-norm
‘fro’	Frobenius norm	–
‘nuc’	nuclear norm	–
inf	max(sum(abs(x), axis=1))	max(abs(x))
-inf	min(sum(abs(x), axis=1))	min(abs(x))
0	–	sum(x != 0)
1	max(sum(abs(x), axis=0))	as below
-1	min(sum(abs(x), axis=0))	as below
2	2-norm (largest sing. value)	as below
-2	smallest singular value	as below
other	–	sum(abs(x)ord)(1./ord)