  We study the joint probability density of the eigenvalues of a product of
rectangular real, complex or quaternion random matrices in a unified way. The
random matrices are distributed according to arbitrary probability densities,
whose only restriction is the invariance under left and right multiplication by
orthogonal, unitary or unitary symplectic matrices, respectively. We show that
a product of rectangular matrices is statistically equivalent to a product of
square matrices. Hereby we prove a weak commutation relation of the random
matrices at finite matrix sizes, which previously have been discussed for
infinite matrix size. Moreover we derive the joint probability densities of the
eigenvalues. To illustrate our results we apply them to a product of random
matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed
version thereof. For these weights we show that the product of complex random
matrices yield a determinantal point process, while the real and quaternion
matrix ensembles correspond to Pfaffian point processes. Our results are
visualized by numerical simulations. Furthermore, we present an application to
a transport on a closed, disordered chain coupled to a particle bath.
