Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems
Sylvester's law of inertia states that the number of positive, negative and zero eigenvalues of Hermitian matrices is preserved under congruence transformations. The same is true of generalized Hermitian definite eigenvalue problems, in which the two matrices are allowed to undergo different co...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Elsevier
2019
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| _version_ | 1797099372650954752 |
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| author | Nakatsukasa, Y Noferini, V |
| author_facet | Nakatsukasa, Y Noferini, V |
| author_sort | Nakatsukasa, Y |
| collection | OXFORD |
| description | Sylvester's law of inertia states that the number of positive, negative and zero eigenvalues of Hermitian matrices is preserved under congruence transformations. The same is true of generalized Hermitian definite eigenvalue problems, in which the two matrices are allowed to undergo different congruence transformations, but not for the indefinite case. In this paper we investigate the possible change in inertia under congruence for generalized Hermitian indefinite eigenproblems, and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil, especially when one of the matrices is almost definite. A prominent application of the original Sylvester's law is in finding the number of eigenvalues in an interval. Our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems. |
| first_indexed | 2024-03-07T05:22:45Z |
| format | Journal article |
| id | oxford-uuid:df7c186b-9ee6-46c2-b596-fa9280c79b1a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T05:22:45Z |
| publishDate | 2019 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:df7c186b-9ee6-46c2-b596-fa9280c79b1a2022-03-27T09:39:40ZInertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problemsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:df7c186b-9ee6-46c2-b596-fa9280c79b1aEnglishSymplectic Elements at OxfordElsevier2019Nakatsukasa, YNoferini, VSylvester's law of inertia states that the number of positive, negative and zero eigenvalues of Hermitian matrices is preserved under congruence transformations. The same is true of generalized Hermitian definite eigenvalue problems, in which the two matrices are allowed to undergo different congruence transformations, but not for the indefinite case. In this paper we investigate the possible change in inertia under congruence for generalized Hermitian indefinite eigenproblems, and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil, especially when one of the matrices is almost definite. A prominent application of the original Sylvester's law is in finding the number of eigenvalues in an interval. Our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems. |
| spellingShingle | Nakatsukasa, Y Noferini, V Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems |
| title | Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems |
| title_full | Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems |
| title_fullStr | Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems |
| title_full_unstemmed | Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems |
| title_short | Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems |
| title_sort | inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems |
| work_keys_str_mv | AT nakatsukasay inertialawsandlocalizationofrealeigenvaluesforgeneralizedindefiniteeigenvalueproblems AT noferiniv inertialawsandlocalizationofrealeigenvaluesforgeneralizedindefiniteeigenvalueproblems |