Suppose $A,B$ are $n\times n$ positive definite real symmetric matrices,P is an $n\times n$ real matrix, prove that $A>P^{T}B^{-1}P$ iff $B>PA^{-1}P^{T}$.
By using the orthogonal diagonalization, I can prove this when $P$ is nonsingular. I have tried Polar decomposition, but I didn't know how to deal with $P$, since it can be singular.