Perron matrix semigroups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916005630967808 |
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| author | Protasov, Vladimir Yu. |
| author_facet | Protasov, Vladimir Yu. |
| contents | We consider multiplicative semigroups of real dxd matrices. A semigroup S is called Perron if each of its matrices has a Perron eigenvalue, i.e., an eigenvalue equal to the spectral radius. If all matrices of S leave a proper convex cone invariant, then S is Perron. Our main result asserts the converse: every irreducible Perron semigroup possesses a common invariant cone, provided that some mild assumptions are satisfied. This gives conditions for a set of matrices to share a common invariant cone, which is an important property widely studied in the literature. Then we address the problem to characterize the exceptions, when a Perron semigroup does not have an invariant cone. For d\le 4, all Perron semigroups are classified. For higher dimensions~$d$, several classes of such semigroups are found. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10571 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perron matrix semigroups Protasov, Vladimir Yu. Rings and Algebras Spectral Theory We consider multiplicative semigroups of real dxd matrices. A semigroup S is called Perron if each of its matrices has a Perron eigenvalue, i.e., an eigenvalue equal to the spectral radius. If all matrices of S leave a proper convex cone invariant, then S is Perron. Our main result asserts the converse: every irreducible Perron semigroup possesses a common invariant cone, provided that some mild assumptions are satisfied. This gives conditions for a set of matrices to share a common invariant cone, which is an important property widely studied in the literature. Then we address the problem to characterize the exceptions, when a Perron semigroup does not have an invariant cone. For d\le 4, all Perron semigroups are classified. For higher dimensions~$d$, several classes of such semigroups are found. |
| title | Perron matrix semigroups |
| topic | Rings and Algebras Spectral Theory |
| url | https://arxiv.org/abs/2502.10571 |