Python dealing with repeated eigenvalues

Question

Consider A a real symmetric matrix and

import scipy
(s,u)=scipy.linalg.eigh(A)

If A has repeated eigenvalues then the columns of u are not necessarily orthonormal. What is the most efficient way to obtain a basis of orthonormal eigenvectors in python?

Answer 1

Use eigh() instead of eig() , since eigh() is specially designed to deal with complex hermitian and real symmetric matrices.

In [1]: import numpy as np
...: sigma = np.array([[ 7, -3, 2],
...:                  [ -3, -1, 6],
...:                  [  2,  6, 4]])

In [2]: val, vec = np.linalg.eig(sigma)

In [3]: val
Out[3]: array([ 8., -6.,  8.])

In [4]: vec
Out[4]: 
array([[ 0.96362411, -0.26726124, -0.06680865],
       [-0.22237479, -0.80178373,  0.56878282],
       [ 0.14824986,  0.53452248,  0.81976991]])

Observe that vec.T @ vec is not an identity matrix:

In [5]: vec.T @ vec
Out[5]: 
array([[ 1.00000000e+00,  0.00000000e+00, -6.93306162e-02],
       [ 0.00000000e+00,  1.00000000e+00,  1.11022302e-16],
       [-6.93306162e-02,  1.11022302e-16,  1.00000000e+00]])

Now let's use eigh() :

In [6]: val, vec = np.linalg.eigh(sigma)

In [7]: val
Out[7]: array([-6.,  8.,  8.])

In [8]: vec
Out[8]: 
array([[-0.26726124,  0.87287156, -0.40824829],
       [-0.80178373,  0.03357198,  0.59667058],
       [ 0.53452248,  0.48679376,  0.69088172]])

Observe that vec.T @ vec is an identity matrix:

In [9]: vec.T @ vec
Out[9]: 
array([[1.00000000e+00, 5.78786501e-17, 5.56940743e-17],
       [5.78786501e-17, 1.00000000e+00, 4.61864879e-17],
       [5.56940743e-17, 4.61864879e-17, 1.00000000e+00]])