On generalized eigenvalues of MAX matrices to MIN matrices and of LCM matrices to GCD matrices
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866909998921023488 |
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| author | Merikoski, Jorma K. Haukkanen, Pentti Sasaki, Antonio Tossavainen, Timo |
| author_facet | Merikoski, Jorma K. Haukkanen, Pentti Sasaki, Antonio Tossavainen, Timo |
| contents | We determine, for any n $\ge$ 1, the generalized eigenvalues of an n x n MAX matrix to the corresponding MIN matrix. We also show that a similar result holds for the generalized eigenvalues of an nxn LCM matrix to the corresponding GCD matrix when n $\le$ 4, but breaks down for n > 4. In addition, we prove Cauchy's interlacing theorem for generalized eigenvalues, and we conjecture an unexpected connection between the OEIS sequence A004754 and the appearance of -1 as a generalized eigenvalue in the LCM-GCD setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16626 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On generalized eigenvalues of MAX matrices to MIN matrices and of LCM matrices to GCD matrices Merikoski, Jorma K. Haukkanen, Pentti Sasaki, Antonio Tossavainen, Timo Number Theory Spectral Theory We determine, for any n $\ge$ 1, the generalized eigenvalues of an n x n MAX matrix to the corresponding MIN matrix. We also show that a similar result holds for the generalized eigenvalues of an nxn LCM matrix to the corresponding GCD matrix when n $\le$ 4, but breaks down for n > 4. In addition, we prove Cauchy's interlacing theorem for generalized eigenvalues, and we conjecture an unexpected connection between the OEIS sequence A004754 and the appearance of -1 as a generalized eigenvalue in the LCM-GCD setting. |
| title | On generalized eigenvalues of MAX matrices to MIN matrices and of LCM matrices to GCD matrices |
| topic | Number Theory Spectral Theory |
| url | https://arxiv.org/abs/2601.16626 |