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2012.118: Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach

2012.118: Yuji Nakatsukasa, Vanni Noferini and Alex Townsend (2012) Vector spaces of linearizations for matrix polynomials: a bivariate polynomial approach.

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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 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2012.118
Deposited By:Yuji Nakatsukasa
Deposited On:24 March 2015

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