On the relation between additive and multiplicative decompositions of rational matrix functions

In this paper, a general frame to study relations between additive and multiplicative representations of rational matrix functions is presented. Various extensions of the positive real lemma, as well as of other classical factorization results, both in the continuous and discrete-time cases, are established. In the case of square factorizations, the map between solutions to an asymmetric algebraic Riccati equation and pairs of factors is shown to be a homeomorphism. In this framework, we also derive a geometric characterization of non-square factorizations.