Let M = P*D, where P is a permutation matrix and D diagonal. If P is also symmetric, then does M have all real eigenvalues?
How about $M = \begin{pmatrix}0&1\\\1&0\end{pmatrix}\begin{pmatrix}1&0\\\0&1\end{pmatrix}$? 


No. If $P$ is the matrix of a transposition (2 by 2) and $D$ is $diag(1, 1)$ the eigenvlues are imaginary. 

