numpy seems return wrong eigenvector for circular matrix

Question

I am using numpy to calculate the eigenvalue and eigenvector of a circular matrix. Here is my code(Hji for j=1,2...6 is predefined):

>>> 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]])

The eigenvalues are correct. However, for the eigenvector, I found out they are not linearly independent

>>> 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]])
>>> 

You can see there are off-diagonal terms(check V[2,5] = -1.69601044e-01) which means they are not linear independent vectors. Since this is a Hermitian matrix, how can its eigenvectors become dependent?

By the way, I also use matlab to calculate it and it returns the right value

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

Answer 1

The result returned by eig is perfectly fine. This can be seen by

np.allclose(v.dot(np.diag(w)).dot(LA.inv(v)),H)

True

Note that the output of eig corresponds to a factorization of the input matrix of the form v * diag(w) * inv(v), which holds for generic diagonalizable matrices. Since eig treats H as having no special structure the returned eigenvectors are not expected to have a special structure, e.g., orthogonal. (Do not confuse orthogonality with linear independence - the columns of v are indeed linearly independent as can be simply verified by the non-zero LA.det(v).)

Function eigh knows that the input matrix is hermitian and returns a more convenient, i.e., orthogonal, set of eigenvectors.