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Matrix $M=P^TP-P-P^T+I$, where $P$ is a probability transition matrix and not symmetric. The elements of $P$ are non-negative and the sum of each column of $P$ is 1. I am wondering whether $M$ is positive semi-definite or not. I tried some random $M$s and all their eigenvalues are non-negative, so I believe the claim maybe true. Please help me to prove or disprove my claim. Thanks a lot:)

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It has nothing to do with transition matrix. Just note that $M=(P-I)^T(P-I)$.

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