Let $K_n$ be the set of all nonsingular $n\times n$ lower triangular $(0,1)$-matrices. Hong and Loewy (2004) introduced the numbers$c_n=\min \{\lambda |\lambda \text{is an eigenvalue of}X{X}^{\mathrm{T}},X\in K_n\},n\in {\mathbb{Z}}_{+}.$ A related family of numbers was considered by Ilmonen, Haukkanen, and Merikoski (2008):$C_n=\max \{\lambda |\lambda \text{is an eigenvalue of}X{X}^{\mathrm{T}},X\in K_n\},n\in {\mathbb{Z}}_{+}.$ 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$c_n\ge \begin{array}{c}\hfill \frac{1}{\sqrt{\frac{1}{25}{\phi }^{-4n}+\frac{2}{25}{\phi }^{-2n}-\frac{2}{5\sqrt{5}}n{\phi }^{-2n}-\frac{23}{25}+n+\frac{2}{25}{\phi }^{2n}+\frac{2}{5\sqrt{5}}n{\phi }^{2n}+\frac{1}{25}{\phi }^{4n}}}\end{array},$ and for n even, one has$c_n\ge \begin{array}{c}\hfill \frac{1}{\sqrt{\frac{1}{25}{\phi }^{-4n}+\frac{4}{25}{\phi }^{-2n}-\frac{2}{5\sqrt{5}}n{\phi }^{-2n}-\frac{2}{5}+n+\frac{4}{25}{\phi }^{2n}+\frac{2}{5\sqrt{5}}n{\phi }^{2n}+\frac{1}{25}{\phi }^{4n}}}\end{array},$ where φ denotes the golden ratio. These lower bounds improve the estimates derived previously by Mattila (2015) and Altınışık et al. (2016). The sharpness of these lower bounds is assessed numerically and it is conjectured that $c_n\sim 5{\phi }^{-2n}$ as $n\to \infty$. In addition, a new closed form expression is derived for the numbers $C_n$, viz.$C_n=\frac{1}{4}{\csc }^2\left(\frac{\pi }{4n+2}\right)=\frac{4n^2}{{\pi }^2}+\frac{4n}{{\pi }^2}+(\frac{1}{12}+\frac{1}{{\pi }^2})+\mathcal{O}\left(\frac{1}{n^2}\right),n\in {\mathbb{Z}}_{+}.$

让 ${ķ}_{ñ}$ 是所有非奇异的集合 $ñ\times ñ$ 下三角 $（0，\mathrm{1个}）$-矩阵。Hong和Loewy（2004）介绍了数字$C_{ñ}=\mathrm{分}\{\lambda |\lambda \text{是的特征值}X{X}^{\mathrm{Ť}}，X\in {ķ}_{ñ}\}，ñ\in {\mathbb{ž}}_{+}。$ Ilmonen，Haukkanen和Merikoski（2008）考虑了一个相关的数字族：$C_{ñ}=\mathrm{最高}\{\lambda |\lambda \text{是的特征值}X{X}^{\mathrm{Ť}}，X\in {ķ}_{ñ}\}，ñ\in {\mathbb{ž}}_{+}。$ 这些数字可用于限制属于的矩阵的奇异值 ${ķ}_{ñ}$并且它们出现在例如幂GCD矩阵的特征值范围，晶格理论的满足和联接矩阵以及相关的数论矩阵中。本文表明，对于n个奇数，一个具有下界$C_{ñ}\ge \begin{array}{c}\hfill \frac{\mathrm{1个}}{\sqrt{\frac{\mathrm{1个}}{25}{\phi }^{-4ñ}+\frac{2}{25}{\phi }^{-2ñ}-\frac{2}{5\sqrt{5}}ñ{\phi }^{-2ñ}-\frac{23}{25}+ñ+\frac{2}{25}{\phi }^{2ñ}+\frac{2}{5\sqrt{5}}ñ{\phi }^{2ñ}+\frac{\mathrm{1个}}{25}{\phi }^{4ñ}}}\end{array}，$和ñ甚至，一个有$C_{ñ}\ge \begin{array}{c}\hfill \frac{\mathrm{1个}}{\sqrt{\frac{\mathrm{1个}}{25}{\phi }^{-4ñ}+\frac{4}{25}{\phi }^{-2ñ}-\frac{2}{5\sqrt{5}}ñ{\phi }^{-2ñ}-\frac{2}{5}+ñ+\frac{4}{25}{\phi }^{2ñ}+\frac{2}{5\sqrt{5}}ñ{\phi }^{2ñ}+\frac{\mathrm{1个}}{25}{\phi }^{4ñ}}}\end{array}，$其中φ表示黄金分割率。这些下界改善了Mattila（2015）和Altınışık等人先前得出的估计。（2016）。这些下界的锐度通过数字评估，并且推测$C_{ñ}〜5{\phi }^{-2ñ}$ 如 $ñ\to \infty$。此外，还为数字导出了一个新的封闭形式表达式$C_{ñ}$，即$C_{ñ}=\frac{\mathrm{1个}}{4}{\csc }^2（\frac{\pi }{4ñ+2}）=\frac{4{ñ}^2}{{\pi }^2}+\frac{4ñ}{{\pi }^2}+（\frac{\mathrm{1个}}{12}+\frac{\mathrm{1个}}{{\pi }^2}）+\mathcal{Ø}（\frac{\mathrm{1个}}{{ñ}^2}），ñ\in {\mathbb{ž}}_{+}。$