Given the product of a unitary matrix and an orthogonal matrix, can it be easily inverted _without_ knowing these factors?

Question

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 $M^{-1}$?

Answer 1

Another direction, conceptually the same: $$MM^T = U\overbrace{OO^T}^IU^T = UU^T$$ $$\Rightarrow (MM^T)^{-1} = U^*U^\dagger = (MM^T)^* = M^*M^\dagger$$ $$\Rightarrow MM^TM^*M^\dagger=I$$

Answer 2

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

$$M^{-1} = M^T M^* M^\dagger$$