I'm trying to show that \begin{equation} P^H ( I_M + PBP^H) ^{-1} P = \big( (P^H P)^{-1} + B \big)^{-1}, \end{equation} where $P$ is an $M$-by-$N$ matrix, $I_M$ is the $M$-by-$M$ identity matrix, $B$ is an $N$-by-$N$ matrix, and $(P^H P)$ is invertible.
I've used various versions of matrix inversion lemmas, but I'm stuck.
How can the above equality be shown?