使用Python的Operator库计算矩阵的特征值和特征向量，以及使用NumPy进行特征值计算的方法

import numpy as np
# 创建一个对角矩阵！
x = np.diag((1, 2, 3))  
print(x)
print(np.linalg.eigvals(x))
a, b = np.linalg.eig(x)  
# 特征值保存在a中，特征向量保存在b中
print(a)
print(b)
# 检验特征值与特征向量是否正确
for i in range(3): 
    if np.allclose(a[i] * b[:, i], np.dot(x, b[:, i])):
        print('Right')
    else:
        print('Error')

[[1 0 0]
 [0 2 0]
 [0 0 3]]
[1. 2. 3.]
[1. 2. 3.]
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
Right
Right
Right

import numpy as np

A = np.array([[1, 1], [1, -2], [2, 1]])
print(A)
# [[ 1  1]
#  [ 1 -2]
#  [ 2  1]]

u, s, vh = np.linalg.svd(A, full_matrices=False)
print(u.shape)  # (3, 2)
print(u)
# [[-5.34522484e-01 -1.11022302e-16]
#  [ 2.67261242e-01 -9.48683298e-01]
#  [-8.01783726e-01 -3.16227766e-01]]

print(s.shape)  # (2,)
print(np.diag(s))
# [[2.64575131 0.        ]
#  [0.         2.23606798]]

print(vh.shape)  # (2, 2)
print(vh)
# [[-0.70710678 -0.70710678]
#  [-0.70710678  0.70710678]]

a = np.dot(u, np.diag(s))
a = np.dot(a, vh)
print(a)
# [[ 1.  1.]
#  [ 1. -2.]
#  [ 2.  1.]]

x = np.array([[1, 2], [3, 4]])
print(x)
# [[1 2]
#  [3 4]]

print(np.linalg.det(x))
# -2.0000000000000004

## 解方程
### 逆矩阵(inverse matrix)
- numpy.linalg.inv(a) 计算矩阵a的逆矩阵（矩阵可逆的充要条件：det(a) != 0，或者a满秩）
import numpy as np

A = np.array([[1, -2, 1], [0, 2, -1], [1, 1, -2]])
print(A)
# [[ 1 -2  1]
#  [ 0  2 -1]
#  [ 1  1 -2]]

# 求A的行列式，不为零则存在逆矩阵
A_det = np.linalg.det(A)  
print(A_det)
# -2.9999999999999996

A_inverse = np.linalg.inv(A)  # 求A的逆矩阵
print(A_inverse)
# [[ 1.00000000e+00  1.00000000e+00 -1.11022302e-16]
#  [ 3.33333333e-01  1.00000000e+00 -3.33333333e-01]
#  [ 6.66666667e-01  1.00000000e+00 -6.66666667e-01]]

x = np.allclose(np.dot(A, A_inverse), np.eye(3))
print(x)  # True
x = np.allclose(np.dot(A_inverse, A), np.eye(3))
print(x)  # True

A_companion = A_inverse * A_det  # 求A的伴随矩阵
print(A_companion)
# [[-3.00000000e+00 -3.00000000e+00  3.33066907e-16]
#  [-1.00000000e+00 -3.00000000e+00  1.00000000e+00]
#  [-2.00000000e+00 -3.00000000e+00  2.00000000e+00]]

#  x + 2y +  z = 7
# 2x -  y + 3z = 7
# 3x +  y + 2z =18

import numpy as np

A = np.array([[1, 2, 1], [2, -1, 3], [3, 1, 2]])
b = np.array([7, 7, 18])
x = np.linalg.solve(A, b)
print(x)  # [ 7.  1. -2.]

x = np.linalg.inv(A).dot(b)
print(x)  # [ 7.  1. -2.]

y = np.allclose(np.dot(A, x), b)
print(y)  # True

a = np.array([1, 2, 3, 4, 5])
b = np.array([4, 5, 6, 7, 8])
print(np.linalg.norm(a-b))

6.708203932499369

x = np.diag([5,5,5,5,5])
print(np.linalg.det(x))
print(np.linalg.inv(x))

3124.999999999999
[[ 0.2  0.   0.   0.   0. ]
 [ 0.   0.2  0.   0.   0. ]
 [ 0.   0.   0.2  0.   0. ]
 [-0.  -0.  -0.   0.2 -0. ]
 [ 0.   0.   0.   0.   0.2]]

import numpy  as np
A=np.array([[1, -2, 1],[0 ,2 ,-8],[-4, 5 ,9]])
b=np.array([0,8,-9])
print(np.linalg.solve(A,b))

[29. 16.  3.]

A=np.array([[4,-1,1],[-1,3,-2],[1,-2,3]])
x,y = np.linalg.eig(A)
print(x)
print(y)

[3. 6. 1.]
[[-8.16496581e-01 -5.77350269e-01  3.89855447e-17]
 [-4.08248290e-01  5.77350269e-01  7.07106781e-01]
 [ 4.08248290e-01 -5.77350269e-01  7.07106781e-01]]