Root vectors of polynomial and rational matrices: theory and computation

TL;DR

This paper extends the concept of root polynomials from polynomial matrices to general rational matrices, providing a theoretical framework and a practical algorithm for eigenvalue and eigenvector computation, even in singular or pole-coalescent cases.

Contribution

It introduces a systematic approach for root vectors of rational matrices, including singular and pole-coalescent cases, with definitions valid over any field and an algorithm for complex matrices.

Findings

Defines eigenvalues and eigenvectors for rational matrices without full rank assumptions.

Provides a practical algorithm using minimal state space realizations and the staircase algorithm.

Studies pole removal techniques for eigenvector recovery in pole cases.

Abstract

The notion of root polynomials of a polynomial matrix $P(\lambda)$ was thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020]. In this paper, we extend such a systematic approach to general rational matrices $R(\lambda)$, possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix $R(\lambda)$, without any need to assume that $R(\lambda)$ has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix $R(\lambda)$ and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point $\lambda_0$. If $\lambda_0$ is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations.