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Suppose I have a projection $P = A(A^T BA)^{-}A^TB$ where $A^{-}$ denotes the Moore-Penrose pseudo-inverse of $A$ and $B$ is symmetric and positive definite. I am wondering, is it true that for any symmetric and positive definite matrix $X$, $Y = PXP^T - X$ is non-negative definite? If so, how would one prove this?

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