Para-Hermitian rational matrices
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913435334213632 |
|---|---|
| author | Dopico, Froilán Noferini, Vanni Quintana, María C. Van Dooren, Paul |
| author_facet | Dopico, Froilán Noferini, Vanni Quintana, María C. Van Dooren, Paul |
| contents | In this paper we study para-Hermitian rational matrices and the associated structured rational eigenvalue problem (REP). Para-Hermitian rational matrices are square rational matrices that are Hermitian for all $z$ on the unit circle that are not poles. REPs are often solved via linearization, that is, using matrix pencils associated to the corresponding rational matrix that preserve the spectral structure. Yet, non-constant polynomial matrices cannot be para-Hermitian. Therefore, given a para-Hermitian rational matrix $R(z)$, we instead construct a $*$-palindromic linearization for $(1+z)R(z)$, whose eigenvalues that are not on the unit circle preserve the symmetries of the zeros and poles of $R(z)$. This task is achieved via Möbius transformations. We also give a constructive method that is based on an additive decomposition into the stable and anti-stable parts of $R(z)$. Analogous results are presented for para-skew-Hermitian rational matrices, i.e., rational matrices that are skew-Hermitian upon evaluation on those points of the unit circle that are not poles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13563 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Para-Hermitian rational matrices Dopico, Froilán Noferini, Vanni Quintana, María C. Van Dooren, Paul Numerical Analysis 65F15, 15A18, 15A22, 15A54, 93B18, 93B20, 93B60 In this paper we study para-Hermitian rational matrices and the associated structured rational eigenvalue problem (REP). Para-Hermitian rational matrices are square rational matrices that are Hermitian for all $z$ on the unit circle that are not poles. REPs are often solved via linearization, that is, using matrix pencils associated to the corresponding rational matrix that preserve the spectral structure. Yet, non-constant polynomial matrices cannot be para-Hermitian. Therefore, given a para-Hermitian rational matrix $R(z)$, we instead construct a $*$-palindromic linearization for $(1+z)R(z)$, whose eigenvalues that are not on the unit circle preserve the symmetries of the zeros and poles of $R(z)$. This task is achieved via Möbius transformations. We also give a constructive method that is based on an additive decomposition into the stable and anti-stable parts of $R(z)$. Analogous results are presented for para-skew-Hermitian rational matrices, i.e., rational matrices that are skew-Hermitian upon evaluation on those points of the unit circle that are not poles. |
| title | Para-Hermitian rational matrices |
| topic | Numerical Analysis 65F15, 15A18, 15A22, 15A54, 93B18, 93B20, 93B60 |
| url | https://arxiv.org/abs/2407.13563 |