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

Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices