On the factorization of rational matrices depending on a parameter

In 1961 Youla published his paper 'On the factorization of rational matrices'. He proved that any proper rational parahermitian matrix, positive definite on the imaginary axis can be factorized as the product of a proper rational matrix, stable with respect to the closed right half plane, and its adjoint. In this paper I prove that for any positive definite, nonstrictly proper matrix this factorization can be given depending analytically on the original matrix, in a sufficiently small neighbourhood. This result is applied to the problem of metrizing the space of transfer matrices of linear systems, in accordance with Vidyasagar's graph topology. Keywords: Graph topology, graph metric, spectral factorization, coprime factorizations, finite dimensional linear systems.