Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach

TL;DR

This paper offers a bivariate polynomial perspective on matrix polynomial linearizations, providing new proofs, generalizations, and insights into their properties, including connections to Bézout matrices and applications to structured polynomials.

Contribution

It introduces a bivariate polynomial approach to linearizations, generalizes the double ansatz space, and explores conditioning for Chebyshev bases, advancing theoretical understanding.

Findings

Every pencil in the double ansatz space relates to a Bézout matrix.

New linearizations are derived with potential applications to structured polynomials.

Analyzed the conditioning of linearizations in Chebyshev basis.

Abstract

We revisit the landmark paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, SIAM J. Matrix Anal. Appl., 28 (2006), pp.~971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key properties of linearizations for matrix polynomials. We also show that every pencil in the double ansatz space is intrinsically connected to a B\'{e}zout matrix, which we use to prove the eigenvalue exclusion theorem. In addition our exposition allows for any polynomial basis and for any field. The new viewpoint also leads to new results. We generalize the double ansatz space by exploiting its algebraic interpretation as a space of B\'{e}zout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials. Moreover, we analyze the conditioning of double ansatz space linearizations in the important practical case of a Chebyshev basis.