A duality relation for matrix pencils with application to linearizations

TL;DR

This paper introduces a new class of stable and well-conditioned linearizations for matrix pencils, based on polynomial algebra and QR factorization, along with a technique to derive new linearizations from existing ones.

Contribution

It presents a novel class of linearizations for matrix pencils, a general technique to generate new linearizations, and connects these to pencil arithmetic for improved analysis.

Findings

The new linearizations have good conditioning and stability properties.

A general technique for deriving linearizations from existing ones is introduced.

The approach extends algebraic operations from matrices to matrix pencils.

Abstract

The aim of this paper is twofold. First, we introduce a new class of linearizations, based on the generalization of a construction used in polynomial algebra to find the zeros of a system of (scalar) polynomial equations. We show that one specific linearization in this class, which is constructed naturally from the QR factorization of the matrix obtained by stacking the coefficients of $A(x)$, has good conditioning and stability properties. Moreover, while analyzing this class, we introduce a general technique to derive new linearizations from existing ones. This technique generalizes some ad-hoc arguments used in dealing with the existing linearization classes, and can hopefully be used to derive a simpler and more general theory of linearizations. This technique relates linearizations to \emph{pencil arithmetic}, a technique used in solving matrix equations that allows to extend some algebraic operations from matrix to matrix pencils.