similarity transformation into symmetric matrices

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?

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The diagonal matrix in question belongs to the center of $GL(n,R)$ and the question itself does not belong to MO. Voting to close. – Misha Jun 4 '13 at 18:21
The matrix $A$ is not necessarily a diagonal matrix. The question may not be central, but still is reasonable. – Dietrich Burde Jun 4 '13 at 19:40
Oh, sorry, I misread the question. – Misha Jun 4 '13 at 21:46

We can say something on such matrices $B$ by characterizing its eigenvalues, which coincide with the eigenvalues of $A$. Since $A$ is a real symmetric matrix, it has real eigenvalues. Hence a necessary condition on $B$ is that it has real eigenvalues. But one can say more. There are necessary and sufficient conditions known for real numbers $\lambda_1,\ldots ,\lambda_r$ to be the eigenvalues of a non-negative symmetric matrix. See for example the paper of M. Fiedler, Eigenvalues of nonnegative symmetric matrices here: http://www.sciencedirect.com/science/article/pii/0024379574900317.
Update: We still have the additional condition that the diagonal of $A$ is $(a,a,\ldots ,a)$, but this is also discussed in the paper of Fiedler. Given $2n$ real numbers $\lambda_i$, one can say whether there is a nonnegative symmetric matrix with eigenvalues $\lambda_1,\ldots ,\lambda_n$ and diagonal $\lambda_{n+1},\ldots ,\lambda_{2n}$.
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).