  Products and sums of random matrices have seen a rapid development in the
past decade due to various analytical techniques available. Two of these are
the harmonic analysis approach and the concept of polynomial ensembles. Very
recently, it has been shown for products of real matrices with anti-symmetric
matrices of even dimension that the traditional harmonic analysis on matrix
groups developed by Harish-Chandra et al. needs to be modified when considering
the group action on general symmetric spaces of matrices. In the present work,
we consider the product of complex random matrices with Hermitian matrices, in
particular the former can be also rectangular while the latter has not to be
positive definite and is considered as a fixed matrix as well as a random
matrix. This generalises an approach for products involving the Gaussian
unitary ensemble (GUE) and circumvents the use there of non-compact group
integrals. We derive the joint probability density function of the real
eigenvalues and, additionally, prove transformation formulas for the
bi-orthogonal functions and kernels.
