Matlab VS Python - eig(A,B) VS sc.linalg.eig(A,B)

Question

I have the following matrices sigma and sigmad:

sigma:

    1.9958   0.7250
    0.7250   1.3167

sigmad:

    4.8889   1.1944
    1.1944   4.2361

If I try to solve the generalized eigenvalue problem in python I obtain:

    d,V = sc.linalg.eig(matrix(sigmad),matrix(sigma))

V:

    -1     -0.5614
    -0.4352    1

If I try to solve the g. e. problem in matlab I obtain:

    [V,d]=eig(sigmad,sigma)

V:

    -0.5897    -0.5278
    -0.2564    0.9400

But the d's do coincide.

Answer 1

Any (nonzero) scalar multiple of an eigenvector will also be an eigenvector; only the direction is meaningful, not the overall normalization. Different routines use different conventions -- often you'll see the magnitude set to 1, or the maximum value set to 1 or -1 -- and some routines don't even bother being internally consistent for performance reasons. Your two different results are multiples of each other:

In [227]: sc = array([[-1., -0.5614], [-0.4352,  1.    ]])

In [228]: ml = array([[-.5897, -0.5278], [-0.2564, 0.94]])

In [229]: sc/ml
Out[229]: 
array([[ 1.69577751,  1.06366048],
       [ 1.69734789,  1.06382979]])

and so they're actually the same eigenvectors. Think of the matrix as an operator which changes a vector: the eigenvectors are the special directions where a vector pointing that way won't be twisted by the matrix, and the eigenvalues are the factors measuring how much the matrix expands or contracts the vector.