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Vector Spaces of Linearizations for Matrix Polynomials: A Bivariate Polynomial Approach

Nakatsukasa, Yuji and Noferini, Vanni and Townsend, Alex (2017) 'Vector Spaces of Linearizations for Matrix Polynomials: A Bivariate Polynomial Approach.' SIAM Journal on Matrix Analysis and Applications, 38 (1). 1 - 29. ISSN 0895-4798

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Abstract

We revisit the landmark paper [D. S. Mackey et al. 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 Bézout 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 Bézout 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.

Item Type: Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Vanni Noferini
Date Deposited: 21 Feb 2017 14:31
Last Modified: 30 Jul 2018 10:15
URI: http://repository.essex.ac.uk/id/eprint/18918

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