Let $M = PD$, where $P$ is a permutation matrix and $D$ diagonal.
If $P$ is also symmetric, then does $M$ have all real eigenvalues?



Eigenvalues of monomial matrices
How about $M = \begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}-1&0\\0&1\end{pmatrix}$?


