Vector spaces of linearizations for matrix polynomials: A bivariate polynomial approach
We revisit the landmark paper [D. S. Mackey et al. 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 doub...
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| Format: | Journal article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2017
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| _version_ | 1797076845287440384 |
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| author | Nakatsukasa, Y Noferini, V Townsend, A |
| author_facet | Nakatsukasa, Y Noferini, V Townsend, A |
| author_sort | Nakatsukasa, Y |
| collection | OXFORD |
| description | We revisit the landmark paper [D. S. Mackey et al. 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 polynomial basis and for any field. The new viewpoint also leads to new results. We generalize the double ansatz space by exploiting its algebraic interpretation as a space of Bézout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials. Moreover, we analyze the conditioning of double ansatz space linearizations in the important practical case of a Chebyshev basis. |
| first_indexed | 2024-03-07T00:09:32Z |
| format | Journal article |
| id | oxford-uuid:78bb767d-40b1-44c7-82f8-4196a064b61a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:09:32Z |
| publishDate | 2017 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:78bb767d-40b1-44c7-82f8-4196a064b61a2022-03-26T20:32:40ZVector spaces of linearizations for matrix polynomials: A bivariate polynomial approachJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:78bb767d-40b1-44c7-82f8-4196a064b61aEnglishSymplectic Elements at OxfordSociety for Industrial and Applied Mathematics2017Nakatsukasa, YNoferini, VTownsend, AWe revisit the landmark paper [D. S. Mackey et al. 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 polynomial basis and for any field. The new viewpoint also leads to new results. We generalize the double ansatz space by exploiting its algebraic interpretation as a space of Bézout pencils to derive new linearizations with potential applications in the theory of structured matrix polynomials. Moreover, we analyze the conditioning of double ansatz space linearizations in the important practical case of a Chebyshev basis. |
| spellingShingle | Nakatsukasa, Y Noferini, V Townsend, A 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 |
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