Toeplitz nonnegative realization of spectra via companion matrices
The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2019-12-01
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| Series: | Special Matrices |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/spma-2019-0017 |
| _version_ | 1831659164172025856 |
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| author | Collao Macarena Salas Mario Soto Ricardo L. |
| author_facet | Collao Macarena Salas Mario Soto Ricardo L. |
| author_sort | Collao Macarena |
| collection | DOAJ |
| description | The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix. |
| first_indexed | 2024-12-19T17:55:24Z |
| format | Article |
| id | doaj.art-20529f2475a34198a3778ef4e281edc7 |
| institution | Directory Open Access Journal |
| issn | 2300-7451 |
| language | English |
| last_indexed | 2024-12-19T17:55:24Z |
| publishDate | 2019-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Special Matrices |
| spelling | doaj.art-20529f2475a34198a3778ef4e281edc72022-12-21T20:11:50ZengDe GruyterSpecial Matrices2300-74512019-12-017123024510.1515/spma-2019-0017spma-2019-0017Toeplitz nonnegative realization of spectra via companion matricesCollao Macarena0Salas Mario1Soto Ricardo L.2Departamento de Matemáticas, Universidad Católica del Norte, Casilla 1280, Antofagasta, ChileDepartamento de Matemáticas, Universidad Católica del Norte, Casilla 1280, Antofagasta, ChileDepartamento de Matemáticas, Universidad Católica del Norte, Casilla 1280, Antofagasta, ChileThe nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.https://doi.org/10.1515/spma-2019-0017toeplitz nonnegative inverse eigenvalue problemunit hessenberg toeplitz matrixsymmetric nonnegative block toeplitz matrixuniversal realizability15a2915a18 |
| spellingShingle | Collao Macarena Salas Mario Soto Ricardo L. Toeplitz nonnegative realization of spectra via companion matrices Special Matrices toeplitz nonnegative inverse eigenvalue problem unit hessenberg toeplitz matrix symmetric nonnegative block toeplitz matrix universal realizability 15a29 15a18 |
| title | Toeplitz nonnegative realization of spectra via companion matrices |
| title_full | Toeplitz nonnegative realization of spectra via companion matrices |
| title_fullStr | Toeplitz nonnegative realization of spectra via companion matrices |
| title_full_unstemmed | Toeplitz nonnegative realization of spectra via companion matrices |
| title_short | Toeplitz nonnegative realization of spectra via companion matrices |
| title_sort | toeplitz nonnegative realization of spectra via companion matrices |
| topic | toeplitz nonnegative inverse eigenvalue problem unit hessenberg toeplitz matrix symmetric nonnegative block toeplitz matrix universal realizability 15a29 15a18 |
| url | https://doi.org/10.1515/spma-2019-0017 |
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