On the maximal spread of symmetric Bohemian matrices
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909854052909056 |
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| author | Calkin, Neil J. Corless, Robert M. Gonzalez-Vega, Laureano Sendra, J. Rafael Sendra, Juana |
| author_facet | Calkin, Neil J. Corless, Robert M. Gonzalez-Vega, Laureano Sendra, J. Rafael Sendra, Juana |
| contents | Let A be a square matrix with real entries. The spread of A is defined as the maximum of the distances among the eigenvalues of A. Let $S_m[a,b]$ denote the set of all $m\times m$ symmetric matrices with entries in the real interval $[a,b]$ and let $S_m\{a,b\}$ be the subset of $S_m[a,b]$ of Bohemian matrices with population from only the extremal elements $\{a,b\}$. S. M. Fallat and J. J. Xing in 2012 proposed the following conjecture: the maximum spread in $S_m[a,b]$ is attained by a rank $2$ matrix in $S_m\{a,b\}$. X. Zhan had proved previously that the conjecture was true for $S_m[-a,a]$ with $a>0$. We will show how to interpret this problem geometrically, via polynomial resultants, in order to be able to treat this conjecture from a computational point of view. This will allow us to prove that this conjecture is true for several formerly open cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15919 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the maximal spread of symmetric Bohemian matrices Calkin, Neil J. Corless, Robert M. Gonzalez-Vega, Laureano Sendra, J. Rafael Sendra, Juana Spectral Theory 05C50, 34L15 Let A be a square matrix with real entries. The spread of A is defined as the maximum of the distances among the eigenvalues of A. Let $S_m[a,b]$ denote the set of all $m\times m$ symmetric matrices with entries in the real interval $[a,b]$ and let $S_m\{a,b\}$ be the subset of $S_m[a,b]$ of Bohemian matrices with population from only the extremal elements $\{a,b\}$. S. M. Fallat and J. J. Xing in 2012 proposed the following conjecture: the maximum spread in $S_m[a,b]$ is attained by a rank $2$ matrix in $S_m\{a,b\}$. X. Zhan had proved previously that the conjecture was true for $S_m[-a,a]$ with $a>0$. We will show how to interpret this problem geometrically, via polynomial resultants, in order to be able to treat this conjecture from a computational point of view. This will allow us to prove that this conjecture is true for several formerly open cases. |
| title | On the maximal spread of symmetric Bohemian matrices |
| topic | Spectral Theory 05C50, 34L15 |
| url | https://arxiv.org/abs/2510.15919 |