Why does the product of the eigenvector and the corresponding eigenvalue not equal the product of the original matrix and the eigenvector?

Question

When I multiply an eigenvector by a matrix, it should result in the same output as multiplying that eigenvector by its corresponding eigenvalue. I am trying to verify that my eigenvectors and eigenvalues are working as advertised, but the outputs don't seem right.

cov_matrix = np.cov(scaled_data)
eig_vals, eig_vecs = np.linalg.eigh(cov_matrix)

a = cov_matrix.dot(eig_vecs[:, 0])
b = eig_vecs[:, 0] * eig_vals[0]

When I print a and b, they are the same shape but their values are all different. What is going wrong here?

Answer 1

Try the following :

import numpy as np

np.random.seed(42) # for reproducibility
A = np.random.random((10,10)) + np.random.random((10,10)) * 1j
eig_vals, eig_vecs = np.linalg.eigh(A)

np.allclose(A @ eig_vecs[:, 0], eig_vals[0] * eig_vecs[:, 0])
>>> False

Keep in mind that np.linalg.eigh return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix. So for a hermitian matrix:

A = (A + A.T.conj())/2  # Here A is forced Hermitian now
eig_vals, eig_vecs = np.linalg.eigh(A)

print(np.allclose(A @ eig_vecs[:, 0], eig_vals[0] * eig_vecs[:, 0]))
>>> True

Check prior to diagonalization if cov_matrix is symmetric with something like np.allclose(cov_matrix, cov_matrix.T.conj()). If not, you can just use np.linalg.eig.