Revisiting the matrix polynomial greatest common divisor

TL;DR

This paper revisits the problem of finding the greatest common right divisor (GCRD) of polynomial matrices, providing new conditions, linking solutions to Smith and Hermite forms, and introducing an efficient algorithm using state-space techniques.

Contribution

It introduces necessary and sufficient conditions for GCRD, characterizes the solution space, and proposes a novel algorithm based on state-space methods and the staircase algorithm.

Findings

Provides a complete characterization of GCRD solutions.

Develops an algorithm using orthogonal transformations and state-space techniques.

Works directly on coefficient matrices with proven effectiveness.

Abstract

In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices $P_i(\lambda)\in \F[\la]^{m_i\times n}$, $i=1,\ldots,k$ with coefficients in a generic field $\F$, and with common column dimension $n$. We give necessary and sufficient conditions for a matrix $G(\la)\in \F[\la]^{\ell\times n}$ to be a GCRD using the Smith normal form of the $m \times n$ compound matrix $P(\lambda)$ obtained by concatenating $P_i(\lambda)$ vertically, where $m=\sum_{i=1}^k m_i$. We also describe the complete set of degrees of freedom for the solution $G(\la)$, and we link it to the Smith form and Hermite form of $P(\la)$. We then give an algorithm for constructing a particular minimum rank solution for this problem when $\F=\C$ or $\R$, using state-space techniques. This new method works directly on the coefficient matrices of $P(\la)$, using orthogonal transformations only. The method is based on the staircase algorithm, applied to a particular pencil derived from a generalized state-space model of $P(\la)$.