NumPy: difference between linalg.eig() and linalg.eigh()

Question

In a Python 3 application I'm using NumPy to calculate eigenvalues and eigenvectors of a symmetric real matrix.

Here's my demo code:

import numpy as np a = np.random.rand(3,3)  # generate a random array shaped (3,3)  a = (a + a.T)/2  # a becomes a random simmetric matrix      evalues1, evectors1 = np.linalg.eig(a)  evalues2, evectors2 = np.linalg.eigh(a) 

Except for the signs, I got the same eigenvectors and eigenvalues using np.linalg.eig and np.linalg.eigh. So, what's the difference between the two methods?

Thanks

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EDIT: I've read the docs here https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.eig.html and here https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.eigh.html but still I can not understand why I should use eigh() when I have a symmetric array.

Answer 1

eigh guarantees you that the eigenvalues are sorted and uses a faster algorithm that takes advantage of the fact that the matrix is symmetric. If you know that your matrix is symmetric, use this function.
Attention, eigh doesn't check if your matrix is indeed symmetric, it by default just takes the lower triangular part of the matrix and assumes that the upper triangular part is defined by the symmetry of the matrix.

eig works for general matrices and therefore uses a slower algorithm, you can check that for example with IPythons magic command %timeit. If you test with larger matrices, you will also see that in general the eigenvalues are not sorted here.