Sometimes the eigenvectors calculated in python and mathematica have opposite signs anyone know why?

Question

I am trying to calculate the eigenvectors of many 3x3 matrices using python. My code in python is based on code from Mathematica which uses the Eigenvector[] function. I have tried using the eig() function from both numpy and scipy and the majority of the time the eigenvectors calculated in mathematica and python are identical. However, there are a few instances which the eigenvectors calculated in python are opposite in sign to those calculated in mathematica.

I have already tried using numpy and scipy to calculate the eigenvectors which both result in the same problem.

Python Code

    _, evec = eig(COVMATRIX)

Mathematica Code

    evec = Eigenvectors[COVMATRIX]

Note that in Mathematica the eigenvectors are along the rows whereas in python the eigenvectors are along the columns in the matrix.

Given

    COVMATRIX = [[2.9296875e-07, 0.0, 2.09676562e-10], 
                 [0.0, 2.9296875e-07, 1.5842226562e-09], 
                 [2.09676562e-10, 1.58422265e-09, 5.85710e-11]]

The eigenvectors from python are

    [[-7.15807155e-04,  9.91354763e-01, -1.31206788e-01],
     [-5.40831983e-03, -1.31208740e-01, -9.91340011e-01],
     [9.99985119e-01,  2.21572611e-13, -5.45548378e-03]]

The eigenvectors from mathematica are

    {{-0.131207, -0.99134, -0.00545548}, 
     {0.991355, -0.131209, 2.6987*10^-13}, 
     {-0.000715807, -0.00540832, 0.999985}}

In this case the eigenvectors are the same between mathematica and python but...

Given

    COVMATRIX = [[2.9296875e-07, 0.0, 6.3368875e-10],
                 [0.0, 2.9296875e-07, 1.113615625e-09],
                 [6.3368875e-10, 1.113615625e-09, 5.0957159954e-11]]

The eigenvectors from python are

    [[ 2.16330513e-03,  8.69137041e-01,  4.94566602e-01],  
     [ 3.80169349e-03, -4.94571334e-01,  8.69128726e-01], 
     [-9.99990434e-01,  1.11146084e-12,  4.37410133e-03]]

But the eigenvectors from python are opposite in sign to the eigenvectors in mathematica which are below

    {{-0.494567, -0.869129, -0.0043741},  
     {0.869137, -0.494571, 1.08198*10^-12}, 
     {-0.00216331, -0.00380169, 0.99999}}

Answer 1

If v is an eigen vector then by definition : Av = lambda * v

So if v is an eigen vector then -v is also an eigen vector since : A * (-v) = - A*v = -lambda * v= lambda * (-v)

So both approaches are correct. The goal of eigen vectors is to find non-colinear vectors (which is helpful if you want to diagonalise your matrix). So it doesn't matter if they give you a vector or the opposite vector.