Suppose $A$ is a real, symmetric, positive matrix, s.t. $P^{t}AP=I$ for some congruention matrix $P$. Prove that $\forall Q\in M_{n}(\mathbb R), Q^{t}AQ=I\iff Q=PU$, where $U$ is orthogonal.
I can't work out how to approach this (the right-to-left, mostly), is this a polar decomposition - can I show that $P$ is symmetric? I can use the fact that $A$ is symmetric to apply unitary diagonalization: $U^{t}AU=D$, but then I'm stuck with a positive $D$... it seems rather easy but I can't come up with the exact arguments.
Let $R = Q^{-1}.$ Then $A = R^T R$ and $P^T R^T R P = I.$ What does that say about $RP \; \; ?$