On some properties of the minimal in degree irreducible factorizations of a rational matrix

The definition and the brief description of the algorithms for constructing the irreducible factorizations of a rational matrix of minimal degree are presented. Two properties of these factorizations are established. First, these factorizations are just the MFDs in application to proper rational matrices and, second, they make it possible to reduce the problem of determining the pole-zero structure of a rational matrix at infinity to the problem of finding the spectral indices of two polynomial matrices at the zero point.