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All inequalities in this post are meant element-wise. Consider a stochastic $n \times n$ matrix $P$, i.e. $P \geq 0, P1 = 1$, where $1$ is a vector of ones. Consider a matrix $A$ of size $n \times m$, $A \geq 0$, whose span includes the row space of $P$. Therefore we have $AA^\dagger P^\top = P^\top$, where $\dagger$ denotes the Moore-Penrose pseudoinverse.

Is it true that $A^\dagger P^\top A \geq 0$ ?

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Consider $$ P=\frac{1}{2}\pmatrix{1&1\\1&1}, \quad A=\pmatrix{1&2&3\\1&1&1}, \quad A^\dagger P^TA=\frac{1}{12}\pmatrix{10&15&20\\4&6&8\\-2&-3&-4}\not\geq 0. $$

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