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| Formato: | Preprint |
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2025
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| Acceso en línea: | https://arxiv.org/abs/2503.14180 |
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| _version_ | 1866915432983691264 |
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| author | Kaarnioja, Vesa Zepernick, André-Alexander |
| author_facet | Kaarnioja, Vesa Zepernick, André-Alexander |
| contents | Let $K_n$ denote the set of all nonsingular $n\times n$ lower triangular $(0,1)$-matrices. Hong and Loewy (2004) introduced the number sequence $$ c_n=\min\{λ\midλ~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad n\in\mathbb Z_+. $$ There have been a number of attempts in the literature to obtain bounds on the numbers $c_n$ by Mattila (2015), Altinisik et al. (2016), Kaarnioja (2021), Loewy (2021), and Altinisik (2021). In this paper, improved upper and lower bounds are derived for the numbers $c_n$. By considering the characteristic polynomial corresponding to the matrix $Z_n$ satisfying $c_n=\|Z_n\|_2^{-1}$, it is shown that the second largest eigenvalue of $Z_n$ is bounded from above by $\frac45$ leading to an improved upper bound on $c_n$. On the other hand, Samuelson's inequality applied to the roots of the characteristic polynomial of $Z_n$ yields an improved lower bound. Numerical experiments demonstrate the quality of the new bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14180 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New upper and lower bounds on the smallest singular values of nonsingular lower triangular $(0,1)$-matrices Kaarnioja, Vesa Zepernick, André-Alexander Combinatorics 06A12, 15B34, 11C20 Let $K_n$ denote the set of all nonsingular $n\times n$ lower triangular $(0,1)$-matrices. Hong and Loewy (2004) introduced the number sequence $$ c_n=\min\{λ\midλ~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad n\in\mathbb Z_+. $$ There have been a number of attempts in the literature to obtain bounds on the numbers $c_n$ by Mattila (2015), Altinisik et al. (2016), Kaarnioja (2021), Loewy (2021), and Altinisik (2021). In this paper, improved upper and lower bounds are derived for the numbers $c_n$. By considering the characteristic polynomial corresponding to the matrix $Z_n$ satisfying $c_n=\|Z_n\|_2^{-1}$, it is shown that the second largest eigenvalue of $Z_n$ is bounded from above by $\frac45$ leading to an improved upper bound on $c_n$. On the other hand, Samuelson's inequality applied to the roots of the characteristic polynomial of $Z_n$ yields an improved lower bound. Numerical experiments demonstrate the quality of the new bounds. |
| title | New upper and lower bounds on the smallest singular values of nonsingular lower triangular $(0,1)$-matrices |
| topic | Combinatorics 06A12, 15B34, 11C20 |
| url | https://arxiv.org/abs/2503.14180 |