为什么对称矩阵的Numpy特征向量不能构造原始矩阵

Question

For a symmetric real matrix A , it can be decomposed as A=Q'UQ , where Q is eigenvectors, U is eigenvalues matrix, Q' is transposed matrix of Q . However, when I use numpy.linalg.eig() to calculate eigenvalues and eigenvectors, for some cases, the result is right, while for some others, it is wrong. For exmaple:

1. A = [[3, -1, -1, -1], [-1, 3, -1, -1], [-1, -1, 3, -1], [-1, -1, -1, 3]]
2. A = [[1, 0, -1, 0], [0, 1, -1, 0], [-1, -1, 3, -1], [0, 0, -1, 1]]

In the Case1, the orginal matrix A can be reconstructed successfully, but in the Case2 the reconstruction is failed. For the second case matrix, I calculate the eigenvalues and eigenvector by hand. The result is right, shown as bellow. I really wonder why?!

The experimental code is as follow:

import numpy as np
import scipy.linalg as spl

N = 4
# case 1
# A = np.array([[3, -1, -1, -1], [-1, 3, -1, -1], [-1, -1, 3, -1], [-1, -1, -1, 3]])
# case 2
A = np.array([[1, 0, -1, 0], [0, 1, -1, 0], [-1, -1, 3, -1], [0, 0, -1, 1]])
lam, vec = np.linalg.eig(A)

# calculate the orthonormal eigenvectors matrix Q
vec = spl.orth(vec)

# orthonormal eigenvectors matrix Q calculated by hand in case 2
# vec = np.array([[np.sqrt(12)/12, np.sqrt(12)/12, -3*np.sqrt(12)/12, np.sqrt(12)/12], [np.sqrt(4)/4, np.sqrt(4)/4, np.sqrt(4)/4, np.sqrt(4)/4], [-np.sqrt(2)/2, np.sqrt(2)/2, 0, 0], [-np.sqrt(6)/6, -np.sqrt(6)/6, 0, 2*np.sqrt(6)/6]]).T

# calculate eigenvalues matrix U
lam_matrix = np.zeros((N,N))
i_0 = [i for i in range(N)]
j_0 = [i for i in range(N)]
lam_matrix[i_0, j_0] = lam

# print the experimental result
print('#### Result ####')
print('eigenvalues')
print(lam)
print('eigenvectors')
print(vec)
print('orthogonality of eigenvectors')
print(vec.T.dot(vec))
print('reconstruct the orginal matix')
print(vec.dot(lam_matrix).dot(vec.T))

Answer 1

scipy.linalg.orth construct an orthonormal basis for the range of the input using SVD, it doesn't always promise to return the orthogonal eigenvectors of matrix A .

To compute the orthogonal eigenvalue decomposition, use eigh instead.

import numpy as np
import scipy.linalg as spl

N = 4
# case 1
# A = np.array([[3, -1, -1, -1], [-1, 3, -1, -1], [-1, -1, 3, -1], [-1, -1, -1, 3]])
# case 2
A = np.array([[1, 0, -1, 0], [0, 1, -1, 0], [-1, -1, 3, -1], [0, 0, -1, 1]])
lam, vec = np.linalg.eigh(A)

# calculate the orthonormal eigenvectors matrix Q
#vec = spl.orth(vec)

# orthonormal eigenvectors matrix Q calculated by hand in case 2
# vec = np.array([[np.sqrt(12)/12, np.sqrt(12)/12, -3*np.sqrt(12)/12, np.sqrt(12)/12], [np.sqrt(4)/4, np.sqrt(4)/4, np.sqrt(4)/4, np.sqrt(4)/4], [-np.sqrt(2)/2, np.sqrt(2)/2, 0, 0], [-np.sqrt(6)/6, -np.sqrt(6)/6, 0, 2*np.sqrt(6)/6]]).T

# calculate eigenvalues matrix U
lam_matrix = np.zeros((N,N))
i_0 = [i for i in range(N)]
j_0 = [i for i in range(N)]
lam_matrix[i_0, j_0] = lam


# print the experimental result
print('#### Result ####')
print('eigenvalues')
print(lam)
print('eigenvectors')
print(vec)
print('orthogonality of eigenvectors')
print(vec.T.dot(vec))
print('reconstruct the orginal matix')
print(vec.dot(lam_matrix).dot(vec.T))

returns

#### Result ####
eigenvalues
[-2.29037709e-16  1.00000000e+00  1.00000000e+00  4.00000000e+00]
eigenvectors
[[-5.00000000e-01  2.26548862e-01 -7.84437556e-01  2.88675135e-01]
 [-5.00000000e-01 -7.92617282e-01  1.96021708e-01  2.88675135e-01]
 [-5.00000000e-01  1.11022302e-16 -5.55111512e-17 -8.66025404e-01]
 [-5.00000000e-01  5.66068420e-01  5.88415848e-01  2.88675135e-01]]
orthogonality of eigenvectors
[[ 1.00000000e+00 -2.40880415e-17  1.99197095e-16  2.65824870e-16]
 [-2.40880415e-17  1.00000000e+00 -1.37886642e-17  2.08372994e-16]
 [ 1.99197095e-16 -1.37886642e-17  1.00000000e+00 -1.85512875e-16]
 [ 2.65824870e-16  2.08372994e-16 -1.85512875e-16  1.00000000e+00]]
reconstruct the orginal matix
[[ 1.00000000e+00 -4.68812634e-16 -1.00000000e+00 -2.12417609e-16]
 [-4.68812634e-16  1.00000000e+00 -1.00000000e+00 -5.88422541e-16]
 [-1.00000000e+00 -1.00000000e+00  3.00000000e+00 -1.00000000e+00]
 [-2.12417609e-16 -5.32911390e-16 -1.00000000e+00  1.00000000e+00]]