eigs finds wrong eigenvectors

[cshaa]

Searching for eigenvalues and eigenvectors of the matrix 1, 2; 3, 4 produces:

math.eigs([
  [1, 2],
  [3, 4]
])
→ {
    values: [-0.3723, 5.372],
    vectors: [
        [1e-17, 2.331],
        [-1.000, 5.097]
    ]
}

while the eigenvalues are correct, the eigenvectors are nonsense. The correct eigenvectors are:

[
   [0.4574, 1],
   [-1.457, 1]
]

[josdejong]

🤔 . Are the vectors always nonsense or just in this specific case?

[cshaa]

I've definitely tested a few matrices after implementing eigs for complex matrices and the results were correct for them. But since I don't know what causes this bug yet, I'm not sure how widespread it is... 🤔

But it is definitely a problem in my code, Arkajit's algorithm for symmetric matrices works OK afaik.

[josdejong]

Maybe there is somewhere a numerical instability causing these issues, I see 1e-17 in the vectors.

[davidtranhq]

I'm looking into this! I brought this bug to m93a's attention and we suspect it may have something to do with the way the Francis algorithm keeps track of transformations.

[cshaa]

@apourzand The behavior you report is correct. The algorithm finds the two eigenvalues, but fails to find eigenvectors – this is fine, as the matrix is a Jordan cell, therefoere not diagonalizable. Sadly, there isn't a way to only run the eigenvalue search without also searching for eigenvectors – for that, see #2180.

[davidtranhq]

Looks like this was fixed with #2478 !

The library now correctly outputs

math.eigs([
  [1, 2],
  [3, 4]
])
→ {
    values: [-0.3723, 5.372],
    vectors: [
        [0.514, 2.331], 
        [-0.353, 5.096]
    ]
}

which are scaled vectors of the correct eigenvectors.

[josdejong]

O that is a nice surprise 😄

I just checked and you're right. I'll close this issue.