Bisymmetric non-negative Jacobi matrix realizations
Within the symmetric inverse eigenvalue problem, the case of bisymmetric Jacobi matrices occupies a central place, since for any strictly monotone list of n real numbers there exists a unique bisymmetric Jacobi matrix realizing the list. Apart from their meaning in several issues such physics, mecha...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/403042 |
| Acceso en línea: | https://hdl.handle.net/2117/403042 https://dx.doi.org/10.1080/03081087.2023.2297391 |
| Access Level: | acceso abierto |
| Palabra clave: | Jacobi matrix Nonnegative matrix Realization Bisymmetric matrix Classificació AMS::15 Linear and multilinear algebra matrix theory Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | Within the symmetric inverse eigenvalue problem, the case of bisymmetric Jacobi matrices occupies a central place, since for any strictly monotone list of n real numbers there exists a unique bisymmetric Jacobi matrix realizing the list. Apart from their meaning in several issues such physics, mechanics, statistics, to cite some of them, the families of this kind of matrices whose spectrum is known are used as models for testing the different algorithms to recover the entries of matrices from spectra data. However, the spectrum is known only for a few families of bisymmetric Jacobi matrices and the examples mainly refer to the case when the spectrum is given by a linear or quadratic function of the order and of the row index. In the first part of the paper, we join all known cases by proving a general result about bisymmetric Jacobi realizations of strictly monotone sequences that are quadratic at most. In the second part we focus on the nonnegative bisymmetric realizations, obtaining new necessary conditions for a given list to be realized by a nonnegative bisymmetric Jacobi matrix. The main novelty in our techniques is considering the gaps between the eigenvalues instead of focusing on the eigenvalues themselves. In the last part of the paper, we explicitly obtain the bisymmetric realization of any list for order less or equal to six. |
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