On symmetric hollow integer matrices with eigenvalues bounded from below
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912195697180672 |
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| author | Jiang, Zilin |
| author_facet | Jiang, Zilin |
| contents | A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, where $ρ$ is the unique real root of $x^3 = x + 1$. We show that for every $λ< λ^*$, there exists $n \in \mathbb{N}$ such that if a symmetric hollow integer matrix has an eigenvalue less than $-λ$, then one of its principal submatrices of order at most $n$ does as well. However, the same conclusion does not hold for any $λ\ge λ^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16860 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On symmetric hollow integer matrices with eigenvalues bounded from below Jiang, Zilin Combinatorics 05C50, 15A18 A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, where $ρ$ is the unique real root of $x^3 = x + 1$. We show that for every $λ< λ^*$, there exists $n \in \mathbb{N}$ such that if a symmetric hollow integer matrix has an eigenvalue less than $-λ$, then one of its principal submatrices of order at most $n$ does as well. However, the same conclusion does not hold for any $λ\ge λ^*$. |
| title | On symmetric hollow integer matrices with eigenvalues bounded from below |
| topic | Combinatorics 05C50, 15A18 |
| url | https://arxiv.org/abs/2408.16860 |