$X$ is $n \times p$ dimension matrix.
$X = PDQ'$ is a thin SVD, where $P$ is $n \times r$, $D$ is $r \times r$, and $Q$ is $p \times r$.
Here is what I tried:
$$(XX')^{-1} = (PDQ'QDP')^{-1} = (PD^2P')^{-1}$$
but since $P$ is just an orthonormal matrix, i.e., $P'P = I_{p}$ but $PP' \neq I_{n}$. How do I pull $P$ out from the inversion?
I'm not sure if it helps to list the source - I encounter this question when I was reading Shao's paper page 814 and 815.