How to prove that a function is unitarily invariant

Question

I am struggling to prove the following, any hints or solutions are very welcome. I have really no idea whee to start. Thank you very much.

Answer 1

If $R$ is a uniformly distributed $n \times n$ orthogonal matrix, then for fixed orthogonal matrices $U$ and $V$ we have $URV$ is also a uniformly distributed orthogonal matrix. Now $\mathrm{det}(xI-U^tAURV^tBVR^t)=\mathrm{det}(U(xI-U^tAURV^tBVR^t)U^t)=\mathrm{det}(xI-AR_0BR_0^t)$, where $R_0=URV^t$ is a uniformly distributed $n\times n$ orthogonal matrix.