论文标题
非词性三角形$（0,1）$ - 矩阵的范围
Bounds on the spectrum of nonsingular triangular $(0,1)$-matrices
论文作者
论文摘要
令$ k_n $为所有非单词$ n \ times n $下三角$（0,1）$ - 矩阵的集合。 Hong and Loewy（2004）引入了数字$$ C_N = {\ rm min} \ {λ\ midλ〜 \ text {是} 〜xx^{\ rm t}的特征， $$由Ilmonen，Haukkanen和Merikoski（2008）考虑了相关的数字家庭：$$ C_N = {\ rm Max} \ {λ\ {λ\ mid λ〜 \ text {是} 〜xx^{\ rm t}的特征n \ in \ mathbb {z} _+。 $$这些数字可用于绑定属于$ k_n $的矩阵的奇异值，例如，在功率GCD矩阵的特征值界中，它们出现，lattice理论相遇和加入矩阵以及相关的数字理论矩阵。在本文中，证明对于$ n $奇数，一个人具有下限$$ C_N \ geq \ frac {1} {\ sqrt {\ frac {1} {25}φ^{ - 4n}+\ frac {2} {25} {25}φ^{ - 2n} - \ frac {2} \ frac {23} {25}+n+\ frac {2} {25}φ^{2n}+\ frac {2} {5 \ sqrt {5}} n或$$，对于$ n $，甚至有$$ C_N \ geq \ frac {1} {\ sqrt {\ frac {1} {25}φ^{ - 4n}+\ frac {4} {25} {25}φ^{ - 2n} - \ frac {2} - \ frac {2} {5}+n+\ frac {4} {25}φ^{2n}+\ frac {2} {5 \ sqrt {5}} n或$$ $φ$表示黄金比率。这些下限改善了Mattila（2015）和Altinişik等人先前得出的估计值。 （2016）。这些下限的清晰度是数值评估的，并猜想$ c_n \ sim5φ^{ - 2n} $作为$ n \ to \ infty $。此外，针对$ c_n $，即的数字派生出了新的封闭式表达式。 $$ C_N = \ frac14 \ csc^2 \ bigG（\fracπ{4n+2} \ bigG）= \ frac {4n^2} {π^2}+\ frac {4n} {π^2}+\ bigg（\ fr ac {1} {12}+\ frac {1} {π^2} \ bigg）+\ \ \ m马理{o} \ bigG（\ frac {1} {n^2} \ bigG），\ quad n \ in \ mathbb {z} _+。 $$
Let $K_n$ be the set of all nonsingular $n\times n$ lower triangular $(0,1)$-matrices. Hong and Loewy (2004) introduced the numbers $$ c_n={\rm min}\{λ\mid λ~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad n\in\mathbb{Z}_+. $$ A related family of numbers was considered by Ilmonen, Haukkanen, and Merikoski (2008): $$ C_n={\rm max}\{λ\mid λ~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad n\in\mathbb{Z}_+. $$ These numbers can be used to bound the singular values of matrices belonging to $K_n$ and they appear, e.g., in eigenvalue bounds for power GCD matrices, lattice-theoretic meet and join matrices, and related number-theoretic matrices. In this paper, it is shown that for $n$ odd, one has the lower bound $$ c_n\geq \frac{1}{\sqrt{\frac{1}{25}φ^{-4n}+\frac{2}{25}φ^{-2n}-\frac{2}{5\sqrt{5}}nφ^{-2n}-\frac{23}{25}+n+\frac{2}{25}φ^{2n}+\frac{2}{5\sqrt{5}}nφ^{2n}+\frac{1}{25}φ^{4n}}}, $$ and for $n$ even, one has $$ c_n\geq \frac{1}{\sqrt{\frac{1}{25}φ^{-4n}+\frac{4}{25}φ^{-2n}-\frac{2}{5\sqrt{5}}nφ^{-2n}-\frac{2}{5}+n+\frac{4}{25}φ^{2n}+\frac{2}{5\sqrt{5}}nφ^{2n}+\frac{1}{25}φ^{4n}}}, $$ where $φ$ denotes the golden ratio. These lower bounds improve the estimates derived previously by Mattila (2015) and Altinişik et al. (2016). The sharpness of these lower bounds is assessed numerically and it is conjectured that $c_n\sim 5φ^{-2n}$ as $n\to\infty$. In addition, a new closed form expression is derived for the numbers $C_n$, viz. $$ C_n=\frac14 \csc^2\bigg(\fracπ{4n+2}\bigg)=\frac{4n^2}{π^2}+\frac{4n}{π^2}+\bigg(\frac{1}{12}+\frac{1}{π^2}\bigg)+\mathcal{O}\bigg(\frac{1}{n^2}\bigg),\quad n\in\mathbb{Z}_+. $$
