Shannon Transform of Certain Matrix Products

We derive a closed form expression for the Shannon transform of the product of two arbitrary positive semidefinite random matrices, where at least one of the matrices is unitary invariant. In principle, S-transforms could be directly applied in this scenario, however in many cases of interest this approach leads to an intractable expression. Our alternative approach, based on hypergeometric functions, leads to a tractable asymptotic expression. This result is validated for finite size matrices using Monte Carlo simulations.