matlab&python中同一个厄米特矩阵的不同特征向量

Question

I'm trying to calculate eigenvalues and eigenvectors of a 3x3 hermitian matrix (named coh). Here is the matlab code I'm using,

coh = [0.327064707875252 + 0.00000000000000i    -0.00770057737827301 + 0.0178948268294334i  -0.00368526462214552 - 0.00615056270163515i
-0.00770057737827302 - 0.0178948268294334i  0.0122797042131420 + 0.00000000000000i  -0.000822583499745789 + 0.000295265015599135i
-0.00368526462214553 + 0.00615056270163516i -0.000822583499745789 - 0.000295265015599135i   0.00526291178539395 + 0.00000000000000i];

[V,D]=eig(coh);

V =

   0.9979 + 0.0000i   0.0229 - 0.0580i   0.0141 + 0.0140i
  -0.0243 - 0.0565i   0.9937 + 0.0000i   0.0936 + 0.0093i
  -0.0114 + 0.0192i  -0.0929 + 0.0104i   0.9954 + 0.0000i

% It should be real valued eigenvalue??
D =

   0.3284 - 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0111 - 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0050 + 0.0000i

I use eig() function in matlab in documentaion it says "When A is real and symmetric or complex Hermitian, the values of e that satisfy Av = λv are real."

To compare the results with python, I used np.linalg.eigh().

import numpy as np
import scipy

coh = np.array([[ 0.32706471+0.j, -0.00770058+0.01789483j,-0.00368526-0.00615056j],
       [-0.00770058-0.01789483j,  0.0122797 +0.j,-0.00082258+0.00029527j],
       [-0.00368526+0.00615056j, -0.00082258-0.00029527j,0.00526291+0.j]])

eigenh = np.linalg.eigh(coh)

The result of eigenvalues and eigen vectors in python are,

%eigenvalue
0.00504925
0.0111318
0.328426

%eigenvector
 (-0.01992713254631731+0.0j)     (0.0623407637085597+0.0j)   (-0.9978559708538679-0.0j)
 (-0.07298515890572027+0.05929161455059334j)     (0.3655270698873978+0.9239915820830416j)    (0.02429370804648004+0.05654205684315627j)
 (-0.706622215529945+0.7010318287578136j)    (-0.043891589739820214-0.08256315733761976j)    (0.011369094995309527-0.01915767907577206j)

There is a significant difference between matlab "eig()" and python "np.linalg.eigh()". I think it is not a normalization problem. Because, when I use the same functions with a symmetric matrix (not hermitian - real valued), the results are exactly the same.

Additional Comment

When I check whether coh matrix (original input as given above) is hermitian or not, matlab returns logical 0.

ishermitian(coh)
ans =
  logical
   0

But, when I round the input matrix by 16, matlab returns logical 1. *Higher rounding value from 16 returns logical 0.

coh2 = round(coh,16)
ishermitian(coh2)
ans =
  logical
   1

Even if I get real valued eigenvalues with rounded input matrix, eigenvectors are still different from python

[V2,D2]=eig(coh2);
V2 =
   0.0141 + 0.0140i   0.0293 - 0.0550i   0.5093 + 0.8581i
   0.0936 + 0.0093i   0.9874 + 0.1110i   0.0362 - 0.0497i
   0.9954 + 0.0000i  -0.0935 + 0.0000i  -0.0223 + 0.0000i

D2 =
    0.0050         0         0
         0    0.0111         0
         0         0    0.3284

Answer 1

If you write imag(D) , you'll see that the imaginary component of the eigenvalues are in the order of 1e-18. That is within rounding error of 0 (when compared to the magnitude of the eigenvalues). But because MATLAB doesn't know the eigenvalues are supposed to be real-valued, it gives them to you as complex numbers.

If you know the eigenvalues are supposed to be real-valued, simply take their real part:

D = real(D);

---

Regarding the edited question:

The matrix coh is almost Hermitian, but not exactly. You can check for example this way:

max(abs(reshape((coh - coh')./coh, 1, [])))

This returns 1.9e-15. This is an order of magnitude larger than eps , which is the reason that MATLAB does not consider it Hermitian.

The matrix entered in the Python code is slightly different. If I copy those values to MATLAB, I see this:

coh2 = [0.32706471+0.0j, -0.00770058+0.01789483j, -0.00368526-0.00615056j
       -0.00770058-0.01789483j,  0.0122797+0.0j, -0.00082258+0.00029527j
       -0.00368526+0.00615056j, -0.00082258-0.00029527j, 0.00526291+0.0j];
max(abs(reshape((coh2 - coh2')./coh2, 1, [])))

This code returns 0. Also ishermitian(coh2) returns true .

Finally, comparing the eigenvalues of coh and coh2 , we find a difference of the order of magnitude 1e-8:

flip(real(eig(coh))) - eig(coh2) % `flip` fixes the ordering difference
ans =
   1.0e-08 *
   0.085548579158157
   0.539922194800480
  -0.238091968363108