In Newman's book Networks, it is given a similarity measure for regular equivalence $\sigma$ (in page 198, 2nd edition and page 217, 1st edition) as a matrix defined by
$\sigma = \alpha A \sigma A$,
where $\alpha$ is a constant and $A$ is the adjacency matrix. Then it is stated:
Although it may not be immediately apparent, this expression is a type of eigenvector equation, where the entire matrix $\sigma$ of similarities is the eigenvector. The parameter $\alpha$ is the eigenvalue (or more correctly, its inverse) [...].
Question: How do you see that $\sigma$ is an eigenvector with $1/\alpha$ the eigenvalue?