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?



Effects of unitarian multiplication into the spectrum of a finite matrix.
By specifying $PP^{\dagger}=W$ you are prescribing the singular values $s\_n$ ($n=1,2,\ldots N$) of the $N\times N$ matrix $P$. These are just the positive square roots of the eigenvalues of the Hermitian, nonnegative matrix $W$. So your question can be rephrased as, what is the relation between the eigenvalues $\lambda\_n$ and the singular values $s\_n$ of the matrix $P$. This is a classic problem studied by Horn (1954), who showed that the only relationships one can state in full generality are those obtained by Weyl (1949):


$\prod\_{n=1}^{k} |\lambda\_n| \leq \prod\_{n=1}^{k} s\_n,$ for $k\lt N$, and
$\prod\_{n=1}^{N} |\lambda\_n| = \prod\_{n=1}^{N} s\_n,$


for the ordering $|\lambda\_1|\gt|\lambda\_2\gt\cdots\gt|\lambda\_N|$ and $s\_1\gt s\_2\gt\cdots\gt s\_N$. (The equality for $k=N$ follows trivially by equating the determinant of $PP^\dagger$ with the determinant of $W$.)


More can be said for random matrices. As shown by Guionnet, Krishnapur, and Zeitouni (arXiv:0909.2214) in a probabilistic sense for “typical matrices”, the singular values almost determine the eigenvalues. In particular, the "single ring theorem" relates the eigenvalue density to the density of singular values.


This [presentation](http://rauhut.ins.uni-bonn.de/WSRMSlides/Rudelson.pdf) by Mark Rudelson gives an introduction.


References:


A. Horn, *On the eigenvalues of a matrix with prescribed singular values*,
Proc. Amer. Math. Soc. **5**, 4–7, (1954).


H. Weyl, *Inequalities between the two kinds of eigenvalues of a linear transformation*, Proc. Nat. Acad. Sci. USA **35**, 408–411, (1949).


