  We use an extension of the diagrammatic rules in random matrix theory to
evaluate spectral properties of finite and infinite products of large complex
matrices and large hermitian matrices. The infinite product case allows us to
define a natural matrix-valued multiplicative diffusion process. In both cases
of hermitian and complex matrices, we observe an emergence of "topological
phase transition" in the spectrum, after some critical diffusion time
$\tau_{\rm crit}$ is reached. In the case of the particular product of two
hermitian ensembles, we observe also an unusual localization-delocalization
phase transition in the spectrum of the considered ensemble. We verify the
analytical formulae obtained in this work by numerical simulation.
