我使用numpy來計算循環矩陣的特徵值和特徵向量。這裏是我的代碼(Hji for j = 1,2 ... 6是預定義的):numpy似乎爲循環矩陣返回錯誤的本徵向量
>>> import numpy as np
>>> H = np.array([H1i, H2i, H3i, H4i, H5i, H6i])
>>> H
array([[ 0., 1., 0., 0., 0., 1.],
[ 1., 0., 1., 0., 0., 0.],
[ 0., 1., 0., 1., 0., 0.],
[ 0., 0., 1., 0., 1., 0.],
[ 0., 0., 0., 1., 0., 1.],
[ 1., 0., 0., 0., 1., 0.]])
>>> from numpy import linalg as LA
>>> w, v = LA.eig(H)
>>> w
array([-2., 2., 1., -1., -1., 1.])
>>> v
array([[ 0.40824829, -0.40824829, -0.57735027, 0.57732307, 0.06604706,
0.09791921],
[-0.40824829, -0.40824829, -0.28867513, -0.29351503, -0.5297411 ,
-0.4437968 ],
[ 0.40824829, -0.40824829, 0.28867513, -0.28380804, 0.46369403,
-0.54171601],
[-0.40824829, -0.40824829, 0.57735027, 0.57732307, 0.06604706,
-0.09791921],
[ 0.40824829, -0.40824829, 0.28867513, -0.29351503, -0.5297411 ,
0.4437968 ],
[-0.40824829, -0.40824829, -0.28867513, -0.28380804, 0.46369403,
0.54171601]])
特徵值是正確的。然而,對於本徵向量,我發現它們不是線性獨立
>>> V = np.zeros((6,6))
>>> for i in range(6):
... for j in range(6):
... V[i,j] = np.dot(v[:,i], v[:,j])
...
>>> V
array([[ 1.00000000e+00, -2.77555756e-17, -2.49800181e-16,
-3.19189120e-16, -1.11022302e-16, 2.77555756e-17],
[ -2.77555756e-17, 1.00000000e+00, -1.24900090e-16,
-1.11022302e-16, -8.32667268e-17, 0.00000000e+00],
[ -2.49800181e-16, -1.24900090e-16, 1.00000000e+00,
-1.52655666e-16, 8.32667268e-17, -1.69601044e-01],
[ -3.19189120e-16, -1.11022302e-16, -1.52655666e-16,
1.00000000e+00, 1.24034735e-01, -8.32667268e-17],
[ -1.11022302e-16, -8.32667268e-17, 8.32667268e-17,
1.24034735e-01, 1.00000000e+00, -1.66533454e-16],
[ 2.77555756e-17, 0.00000000e+00, -1.69601044e-01,
-8.32667268e-17, -1.66533454e-16, 1.00000000e+00]])
>>>
可以看到有非對角線項(查看V [2,5] = -1.69601044e-01),這意味着它們不是線性獨立向量。由於這是一個Hermitian矩陣,它的特徵向量如何變得依賴?
順便說一句,我也用MATLAB來計算的話,它返回正確的價值
V =
0.4082 -0.2887 -0.5000 0.5000 0.2887 -0.4082
-0.4082 -0.2887 0.5000 0.5000 -0.2887 -0.4082
0.4082 0.5774 0 0 -0.5774 -0.4082
-0.4082 -0.2887 -0.5000 -0.5000 -0.2887 -0.4082
0.4082 -0.2887 0.5000 -0.5000 0.2887 -0.4082
-0.4082 0.5774 0 0 0.5774 -0.4082
D =
-2.0000 0 0 0 0 0
0 -1.0000 0 0 0 0
0 0 -1.0000 0 0 0
0 0 0 1.0000 0 0
0 0 0 0 1.0000 0
0 0 0 0 0 2.0000
非對角線項大致爲0.0000000000000001。由於浮點數學的不精確性,它們只是「舍入誤差」。 – BrenBarn
@BrenBarn。對不起,我沒有說清楚,你可以查看V [2,5] = -1.69601044e-01。 – Aaron