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

TL;DR

This paper investigates how the complete eigenstructure of a polynomial matrix transforms under a generic rational change of variable, establishing a general theorem relating their eigenvalues, elementary divisors, and minimal indices.

Contribution

It provides a general theorem that describes the relationship between the eigenstructures of polynomial matrices before and after a rational transformation, covering the most general conditions.

Findings

Eigenvalues and elementary divisors are related through the rational transformation.

Minimal indices of the polynomial matrices are connected via the transformation.

The theorem applies under very general hypotheses, broadening previous results.

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 $Q(y) :=[d(y)]^gP(x(y))$ 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.