30

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

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.

Michael H.
  • 3,323
  • 2
  • 23
  • 31
decadenza
  • 2,380
  • 3
  • 18
  • 31
  • 3
    Read the docs? https://docs.scipy.org/doc/numpy-1.10.1/reference/generated/numpy.linalg.eigh.html – MaxNoe Aug 01 '17 at 10:14
  • if you look at `evectors1` and `evectors2` - it's not just the sign that got flipped for some elements, the position of some elements have changed too. I would go with the read-the-doc option! (if you on in ipython notebook, just run `??np.linalg.eig` and `??np.linalg.eigh` to see the docs! – Atlas7 Aug 01 '17 at 10:18
  • Here is a hint: remove/comment line 4 and try again ;) – Feodoran Aug 01 '17 at 10:27

1 Answers1

59

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.

Michael H.
  • 3,323
  • 2
  • 23
  • 31
  • 2
    Excellent answer! – Kaushal28 May 04 '19 at 09:11
  • Warning: Numpy's documentation of `eig` states that `The eigenvalues are not necessarily ordered.` – kilojoules Nov 25 '19 at 19:29
  • Since `eig` specifically mentions that it returns the **right** eigenvectors, does that imply `eigh` may return the **left** eigenvectors? – Rufus Oct 18 '20 at 15:53
  • I don't think so. According to https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Left_and_right_eigenvectors, "eigenvector" usually refers to "right eigenvector". The documentation for the return values specifies both for `eig` and `eigh` that eigenvectors can be found in columns. – Michael H. Oct 19 '20 at 14:53