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

Nakatsukasa, Yuji and Noferini, Vanni and Townsend, Alex (2012) Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach. [MIMS Preprint]

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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 interpration 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 linearization in the important practical case of a Chebyshev basis.

Item Type: MIMS Preprint
Uncontrolled Keywords: matrix polynomials; bivariate polynomials; B´ezoutian; double ansatz space; degreegraded polynomial basis; orthogonal polynomials; conditioning
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Yuji Nakatsukasa
Date Deposited: 24 Mar 2015
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2276

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