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

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

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Abstract

We revisit the important 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 Bezout matrix, which we use to prove the eigenvalue exclusion theorem. In addition our exposition allows for any degree-graded basis, the monomials being a special case. Matlab code is given to construct the pencils in the double ansatz space for matrix polynomials expressed in any orthogonal basis.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	matrix polynomials, bivariate polynomials, B\'{e}zout matrix, degree-graded basis, structure-preserving linearizations, polynomial eigenvalue problem, matrix pencil
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:	21 Dec 2012
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1933

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• Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach. (deposited 21 Dec 2012) [Currently Displayed]
  • Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach. (deposited 24 Mar 2015)

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