In order to characterize the performances of MIMO systems that depend directly on the distribution of the eigenvalues of random Hermitian matrix so I would like to feature the quality of some particular case that is defined as. For a given random complex gaussian matrix A $(N, M)$ and $B(N, K)$ Gaussian matrix and constant parameter $\alpha$, where ${\tilde A}$ indicate the Moore–Penrose inverse of $A$. The problem consists to find the expectations defined as:

\begin{align} E[\operatorname{trace}((aI + \frac{{\tilde A\,B{B^H}{{\tilde A}^H}}}{{\operatorname{trace}(\tilde A\,B{B^H}{{\tilde A}^H})}})({B^H}B))]\end{align}