TITLE: similarity transformation into symmetric matrices
QUESTION [6 upvotes]: I want to determine some structures of matrices that can be transformed into a symmetric matrices using similarity transformation, i.e.,
$B=T^{-1}AT$
where $T$ is the similarity transformation matrix. 
Here, $A$ is a non-negative matrix and the diagonal elements of $A$ is supposed to be the same $(diag(A)=[a, a, \ldots , a])$.
Any ideas?

REPLY [2 votes]: An addition to Dietrich Burde's answer: Since symmetric matrices are diagonizable, they are semisimple (each invariant subspace has an invariant complement). Thus $B$ has to be semisimple ($\iff$ the minimal polynomial is square free).