Hello everyone,

I am having some difficulties with the Eigen methods for Eigenvalues and Eigenvectors. In particular, I am trying to compute the E-Value and E-Vectors for this matrix:

0 0 1
1 0 0
0 1 0

This is what I get using the Eigen methods:

The eigenvalues of the Adjacency Matrix are:
(-0.5,0.866025)
(-0.5,-0.866025)
(1,0)
The eigenvectors of the Adjacency Matrix are:
(0.0474579,0.575396) (0.0474579,-0.575396) (-0.57735,0)
(0.474579,-0.328798) (0.474579,0.328798) (-0.57735,0)
(-0.522037,-0.246598) (-0.522037,0.246598) (-0.57735,0)

But using Matlab and number of other alternatives, I get a different Eigenvector corresponding to the real eigenvalue, namely positive values in instead:

[ 0,5773502691896258 ; 0,5773502691896258 ; 0,5773502691896258 ]

I actually pulled the example matrix from Wikipedia, where it was used to demonstrate the Perron–Frobenius theorem, which asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components. Thus the vector should be positive for sure. Does someone have an idea why I am getting negative values?

Thank you for your help.

Niccola

I have to admit, I ignored that fact that the negative version of an eigenvector, is still an eigenvector of the corresponding matrix. So I did notice, that Eigen appears to randomly return the negative or positive version of the Eigenvectors. I guess I can just always take the absolute values and work with those.

Indeed, if v is an eigenvector, then -v is also an eigenvector, however, there is no such notion of the absolute value of a vector. For instance, if v=(-1,1), then -v=(1,-1), and there is no reason to prefer one or the other.