The behavior of the complete eigenstructure of a polynomial matrix under a generic rational transformation

Abstract

Given a polynomial matrix P(x) of grade g and a rational function x(y) = n(y)/d(y), where n(y) and d(y) are coprime nonzero scalar polynomials, the polynomial matrix Q(y) := (d(y)) g P(x(y)) is defined. The complete eigenstructures of P(x) and Q(y) are related, including characteristic values, elementary divisors and minimal indices. A theorem on the matter, valid in the most general hypotheses, is stated and proved.