Let $P$ be a stochastic matrix (square, non-negative,rows sum to one). Let $\Xi$ be a diagonal matrix with any left principal eigenvector of $P$ on the diagonal and zeros elsewhere (stationary distribution if $P$ is ergodic).
Prove or give a counterexample for the following.
$\Xi (I - P)$ has eigenvalues with non-negative real parts.
I am not sure if this can be called `positive semi-definite' since the matrix is not symmetric.