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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| Aineistotyyppi: | Report |
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SIMAX
2012
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| _version_ | 1826301794758688768 |
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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. |
| first_indexed | 2024-03-07T05:37:46Z |
| format | Report |
| id | oxford-uuid:e47f9d13-cc08-445f-a7a9-b9e2e64b8b77 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:37:46Z |
| publishDate | 2012 |
| publisher | SIMAX |
| record_format | dspace |
| 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 |