An $m*n$ complex matrix $M$ is given. I take its SVD, $M=U\Sigma V^\dagger$. My question is, are the singular vectors $\textbf{u}_n$ and $\textbf{v}_n$ degenerate? Can I pick any phase for one that will result in some other phase for the other? Or are the singular vector pairs unique?
If so, would it be possible to fix the phase of either the left or the right singular vectors? I would like to decompose $M$ into a real matrix $U$ and a complex one $V$.