I am interested in the following problem: Let $P$ be a $n\times n$ complex finite matrix such as $PP^\dagger =W$. Given $W$, what can I say about the spectrum of $P$?

This matrix "square-root" has of course no unique solution, for if $P$ is a solution, $PU$ is also a solution if $U$ is unitary. If we consider the space $M_n(\mathbb{C})/U(n)$ we can seek an unique answer $P_* $, but I could not determine the shape of the space spanned by the eigenvalues of the equivalence class of $P_*$, apart for $n=2$ and some pretty horrible non-linear equations involving too many variables. Spectral theory is not my strong point, so I was wondering if anyone knew an answer to this: can we say what happens to the eigenvalues of $P$ if we multiply it by $U$ unitary?