Python: np.linalg.eigvalsh returning negatve eigen values

Question

I have a 7x7 covariance matrix (represented as an numpy array).

t = np.array(
    [
        [1.4, 0.3, 0.4, 0.8, 0.4, 0.9, 0.3],
        [0.3, 1.3, 0.4, 2.3, 0.4, 2.4, 0.4],
        [0.4, 0.4, 1.3, 2.8, 0.4, 1.0, 0.3],
        [0.8, 2.3, 2.8, 9.5, 1.0, 7.0, 1.0],
        [0.4, 0.4, 0.4, 1.0, 1.1, 1.2, 0.3],
        [0.9, 2.4, 1.0, 7.0, 1.2, 7.7, 1.0],
        [0.3, 0.4, 0.3, 1.0, 0.3, 1.0, 0.5],
    ],
    dtype=np.float64,
)

I have checked this matrix to be symmetric.

np.allclose(t, t.T)
True

And np.linalg.svd returns valid non-negative singular values. However, np.linalg.eigvalsh, is returning a negative eigenvalue.

min(np.linalg.eigvalsh(t))
-0.06473876145336957

This doesnt make too much sense to me as I have checked that the column of the matrix are linearly independent (getting the reduced row echelon form of the matrix).

import sympy
reduced_form, inds = sympy.Matrix(t.values).rref()

I see similar issues where people have reported eigvalsh to return negative eigenvalue for a well behaved matrix but none of the suggestions have helped. Examples:

I was wondering if anyone has faced a similar issue with np.linalg.eigvalsh and have any recommendations on how to solve it?

Thank you so much.

Have you checked whether things match your expectations when you declare the matrix to be complex128 instead of float64, then, considering these previous questions? If not, what happened? — Marcus Müller, Apr 14 '23 at 06:01
@MarcusMüller i just re-ran the code snippets above with dtype=np.complex128. Unfortunately, I still get a negative eigenvalue, in fact the eigenvalues are unchanged when I use dtype.float64 and dtype.complex128. — vpy, Apr 14 '23 at 06:07
Note that this negative eigenvalue is more than -0.1, that is not that negative. And your "covariance" matrix is obviously rounded to 0.1 precision. In other words, with the precision you have chosen, -0.06 is positive... — chrslg, Apr 14 '23 at 07:20

score 2 · Accepted Answer · answered Apr 14 '23 at 06:18

It's totally normal for symmetric matrices to have negative eigenvalues. A matrix being Hermitian only guarantees that its eigenvalues are real. It doesn't guarantee that the eigenvalues are positive.

It's not normal for a covariance matrix to have negative eigenvalues, meaning this isn't actually a covariance matrix, despite what you thought. Perhaps you rounded the entries of an actual covariance matrix?

Python: np.linalg.eigvalsh returning negatve eigen values

1 Answers1