A duality relation for matrix pencils with application to linearizations
For researchers in numerical linear algebra, this work provides a more stable linearization method and a unifying framework for deriving linearizations, though it is incremental.
The paper introduces a new class of linearizations for matrix pencils based on QR factorization, showing improved conditioning and stability. It also presents a general technique to derive new linearizations from existing ones, generalizing ad-hoc arguments.
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.