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IMA Journal of Numerical Analysis Advance Access originally published online on February 27, 2008
IMA Journal of Numerical Analysis 2009 29(1):141-157; doi:10.1093/imanum/drm051
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Linearization of matrix polynomials expressed in polynomial bases

A. Amiraslani{dagger}

Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

R. M. Corless{ddagger}

Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7

P. Lancaster§

Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

{dagger} Corresponding author. Email: amiram{at}math.ucalgary.ca

{ddagger} Email: rcorless{at}uwo.ca

§ Email: lancaste{at}math.ucalgary.ca

Received on 6 December 2006. Revised on 13 August 2007.


  Abstract

This paper concerns regular matrix polynomials P({lambda}) when represented in various polynomial bases (other than the monomials 1, {lambda}, {lambda}2, ...). As in the monomial case, matrices of ‘companion’ form play an important part in theory and numerical practice. In particular, they are used here to construct ‘strong linearizations’ of P({lambda}). The paper contains three theorems concerning linearizations constructed for representations in a general class of ‘degree-graded’ polynomials, Bernstein polynomials and Lagrange polynomials.

Key Words: matrix polynomials; linearization; orthogonal polynomials


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