验证本征值和本征向量

验证python中根据np.linalg.eig生成的本征值和本征向量的对应关系：

import numpy as np

dim = 4
M = np.random.randn(dim, dim)
M = M + M.T
print(M)

lm, u = np.linalg.eig(M)
print('Eigenvalues:')
print(lm)

print('Eigenvectors:')
print(u)

for i in range(0, 4):
    e_val = lm[i]
    v = []
    for j in range(0, 4):
        e_vec = u[j][i]
        v.append(e_vec)
    a = np.array(v)
    a = a.reshape(-1, 1)  # 将行向量转置为列向量，这里先转换为np.array，再用reshape
    print('M * a:')
    print(M.dot(a))
    print('r * a:')
    print(e_val * a)

import numpy as np

dim = 4
M = np.random.randn(dim, dim)
M = M + M.T
print(M)

lm, u = np.linalg.eig(M)
print('Eigenvalues:')
print(lm)

print('Eigenvectors:')
print(u)

for i in range(0, 4):
    e_val = lm[i]
    a = u[:, i]  # 这种方法可以验证，但是得到的向量是行向量
    print('M * a:')
    print(M.dot(a))
    print('r * a:')
    print(e_val * a)