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

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 s...

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Автори: Townsend, A, Noferini, V, Nakatsukasa, Y
Формат: Report
Опубліковано: SIMAX 2012
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author Townsend, A
Noferini, V
Nakatsukasa, Y
author_facet Townsend, A
Noferini, V
Nakatsukasa, Y
author_sort Townsend, A
collection OXFORD
description 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 Bézout 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.
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spelling oxford-uuid:e47f9d13-cc08-445f-a7a9-b9e2e64b8b772022-03-27T10:17:07ZVector spaces of linearizations for matrix polynomials: A bivariate polynomial approachReporthttp://purl.org/coar/resource_type/c_93fcuuid:e47f9d13-cc08-445f-a7a9-b9e2e64b8b77Mathematical Institute - ePrintsSIMAX2012Townsend, ANoferini, VNakatsukasa, YWe 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 Bézout 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.
spellingShingle Townsend, A
Noferini, V
Nakatsukasa, Y
Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach
title Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach
title_full Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach
title_fullStr Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach
title_full_unstemmed Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach
title_short Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach
title_sort vector spaces of linearizations for matrix polynomials a bivariate polynomial approach
work_keys_str_mv AT townsenda vectorspacesoflinearizationsformatrixpolynomialsabivariatepolynomialapproach
AT noferiniv vectorspacesoflinearizationsformatrixpolynomialsabivariatepolynomialapproach
AT nakatsukasay vectorspacesoflinearizationsformatrixpolynomialsabivariatepolynomialapproach