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$P$ is a stochastic matrix (square, non-negative, rows sum to 1). $\Xi$ is a diagonal matrix with a left principal eigenvector of $P$ on the diagonal and zeros elsewhere (stationary distribution if $P$ is ergodic).

Prove that $\Xi - P^\top \Xi P$ is positive semi-definite.

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Let $u=(1,\ldots,1)$. Notice that $P^t\Xi u=\Xi u$ and $Pu=u$.

Therefore, $P^t\Xi Pu=P^t\Xi u=\Xi u $.

So, $\Xi_{ii}=\sum_{j=1}^n\Xi_{ij}=\sum_{j=1}^n(P^t\Xi P)_{ij}$.

Next, $\Xi_{ii}-(P^t\Xi P)_{ii}=\sum_{j\neq i}(P^t\Xi P)_{ij}=\sum_{j\neq i}|-(P^t\Xi P)_{ij}|\geq 0$.

Therefore, $\Xi-P^t\Xi P$ is diagonally dominant symmetrix matrix with non negative coefficients in the diagonal. Therefore $\Xi-P^t\Xi P$ is positive semidefinite.

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