Given the product of a unitary matrix and an orthogonal matrix, can it be easily inverted _without_ knowing these factors? Given the product $M$ of a unitary matrix $U$ (i.e. $U^\dagger U=1$) and an orthogonal matrix $O$ (i.e. $O^TO=1$), can it be easily inverted without knowing $U$ and $O$?
Sure enough, if $M=UO$, then $M^{-1}=O^TU^\dagger$. But assuming you only know that $M$ is composed in such a way, but not how $U$ and $O$ actually look, does there still exist a simple formula for $M^{-1}$?

Note that
$$M^\dagger M = O^\dagger\underbrace{U^\dagger U}_{=1} O = O^\dagger O$$
Therefore,
$$(M^\dagger M)(M^\dagger M)^T = O^\dagger \underbrace{O O^T}_{=1} O^* = (OO^T)^* = 1$$
So that
$$(M^\dagger)^{-1} = M(M^\dagger M)^T$$
And thus
$$M^{-1} = M^T M^* M^\dagger$$
