I want to compute the average of a product of random matrices:
$$\langle R^\dagger A R \rangle $$
where $R$ and $A$ are uncorrelated $3\times3$ random unitary matrices (with $\dagger$ the conjugate transpose).
After some numerical tests it seems that this simplifies to $\langle A \rangle$. Is this true in general, and is there a simple way to prove it?