Let $P$ be a markov transition matrix, that is, all entries of $P$ satisfies $0\le P_{ij}\le 1$, and row sums equal to $1$.
Do we have the property that: $v^T(I-P)^2v\ge 0$, for any vector $v$ and I a identity matrix? And if the answer is yes, do we also have $v^T(I-P)M(I-P)v\ge 0$ where M is an arbitrary positive semidefinite matrix?
It seems true with some computer simulation, and I didn't find any counter example.
My attempt is to find the spectrum of $(I-P)^2+(I-P^T)^2$, but didn't get anything. Thanks.