Eigenvalue density of correlated complex random Wishart matrices

Using a character expansion method, we calculate exactly the eigenvalue density of random matrices of the form ${\mathbf{M}}^{\ifmmode\dagger\else\textdagger\fi{}}\mathbf{M}$ where $\mathbf{M}$ is a complex matrix drawn from a normalized distribution $P(\mathbf{M})\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}\text{Tr}{\mathbf{A}\mathbf{M}\mathbf{B}{\mathbf{M}}^{\ifmmode\dagger\else\textdagger\fi{}}})$ with $\mathbf{A}$ and $\mathbf{B}$ positive definite (square) matrices of arbitrary dimensions. Such so-called correlated Wishart matrices occur in many fields ranging from information theory to multivariate analysis.