Unexpected Eigenvectors In Numpy

I have seen this question, and it is relevant to my attempt to compute the dominant eigenvector in Python with numPy. I am trying to compute the dominant eigenvector of an n x n ma

Solution 1:

You are just misinterpreting eig's return. According to the docs, the second return argument is

The normalized (unit “length”) eigenvectors, such that the column v[:,i] is the eigenvector corresponding to the eigenvalue w[i].

So the eigenvector corresponding to eigenvalue 1 is not [ 0.4472136 , -0.70710678], but [-0.70710678, -0.70710678], as can be easily verified:

>>>markov.dot([ 0.4472136 , -0.70710678]) # not an eigenvector
array([ 0.21634952, -0.59167474])
>>>markov.dot([-0.70710678, -0.70710678]) # an eigenvector
array([-0.70710678, -0.70710678])