Clever Geek Handbook

Largest Eigenvector and Possible Scipy Weirdness

Not sure if this is a package bug or some other reason, but here we go.

I am using the following package to find the largest eigenvalue and its corresponding eigenvector over a symmetric similarity score matrix (size 10x10):

scipy.sparse.linalg.eigen.arpack.eigsh

So:

scipy.sparse.linalg.eigen.arpack.eigsh(mymatrix, 1, which='LM')

Now my question is, when I run it multiple times (using the same matrix, settings, etc.), sometimes the values ​​in the eigenvectors are positive and sometimes negative (see Run 3 ).

Does anyone know why this might be, or is this a bug? It doesn't seem to have a template, but it only happens when I run the code without closing Python after each iteration (for example, I press F5 after each run).

### Run 1: ###
[[-0.31056873]
[-0.31913092]
[-0.3149287 ]
[-0.32262921]
[-0.32190688]
[-0.31292658]
[-0.32424732]
[-0.31885208]
[-0.31808024]
[-0.298174  ]]

### Run 2: ###
[[-0.31056873]
[-0.31913092]
[-0.3149287 ]
[-0.32262921]
[-0.32190688]
[-0.31292658]
[-0.32424732]
[-0.31885208]
[-0.31808024]
[-0.298174  ]]

### Run 3:###
[[ 0.31056873]
[ 0.31913092]
[ 0.3149287 ]
[ 0.32262921]
[ 0.32190688]
[ 0.31292658]
[ 0.32424732]
[ 0.31885208]
[ 0.31808024]
[ 0.298174  ]]

### Run 4: ###
[[-0.31056873]
[-0.31913092]
[-0.3149287 ]
[-0.32262921]
[-0.32190688]
[-0.31292658]
[-0.32424732]
[-0.31885208]
[-0.31808024]
[-0.298174  ]]

This is not a major problem, but I don't like the vagueness in my code; -)

Thank you very much in advance,

Martyn

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2 answers


This is actually a mathematical question.

But the reason is that there is an arbitrary phase when calculating the eigenvectors. You are solving Ax = bx for x. The equation is invariant under multiplication by (possibly ocmplex) phase.



As for why this is happening (seemingly) randomly, I don't know. But I'm sure this is not a mistake.

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You are using a function with 'LM'

in a which

function parameter . Using these settings, you are looking for the eigenvalue of the highest magnitude , I noticed that the smallest and largest eigenvalues ​​of your matrix are pretty close in absolute value. Add that to the fact that the algorithm evaluates the eigenvalues ​​and you get the answer to your question.
Try to run the code with which = 'LR'

, this will give you your own value with the highest real value (your own values ​​are real anyway).



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