非奇异三角 (0,1)-矩阵谱的界
组合数学
2020-09-23 v2
摘要
设 K n K_n K n n × n n\times n n × n ( 0 , 1 ) (0,1) ( 0 , 1 ) c n = m i n { λ ∣ λ 是 X X T 的特征值 , X ∈ K n } , n ∈ Z + . c_n={\rm min}\{\lambda\mid \lambda~\text{是}~XX^{\rm T}~\text{的特征值},~X\in K_n\},\quad n\in\mathbb{Z}_+. c n = min { λ ∣ λ 是 X X T 的特征值 , X ∈ K n } , n ∈ Z + . C n = m a x { λ ∣ λ 是 X X T 的特征值 , X ∈ K n } , n ∈ Z + . C_n={\rm max}\{\lambda\mid \lambda~\text{是}~XX^{\rm T}~\text{的特征值},~X\in K_n\},\quad n\in\mathbb{Z}_+. C n = max { λ ∣ λ 是 X X T 的特征值 , X ∈ K n } , n ∈ Z + . K n K_n K n n n n c n ≥ 1 1 25 φ − 4 n + 2 25 φ − 2 n − 2 5 5 n φ − 2 n − 23 25 + n + 2 25 φ 2 n + 2 5 5 n φ 2 n + 1 25 φ 4 n , c_n\geq \frac{1}{\sqrt{\frac{1}{25}\varphi^{-4n}+\frac{2}{25}\varphi^{-2n}-\frac{2}{5\sqrt{5}}n\varphi^{-2n}-\frac{23}{25}+n+\frac{2}{25}\varphi^{2n}+\frac{2}{5\sqrt{5}}n\varphi^{2n}+\frac{1}{25}\varphi^{4n}}}, c n ≥ 25 1 φ − 4 n + 25 2 φ − 2 n − 5 5 2 n φ − 2 n − 25 23 + n + 25 2 φ 2 n + 5 5 2 n φ 2 n + 25 1 φ 4 n 1 , n n n c n ≥ 1 1 25 φ − 4 n + 4 25 φ − 2 n − 2 5 5 n φ − 2 n − 2 5 + n + 4 25 φ 2 n + 2 5 5 n φ 2 n + 1 25 φ 4 n , c_n\geq \frac{1}{\sqrt{\frac{1}{25}\varphi^{-4n}+\frac{4}{25}\varphi^{-2n}-\frac{2}{5\sqrt{5}}n\varphi^{-2n}-\frac{2}{5}+n+\frac{4}{25}\varphi^{2n}+\frac{2}{5\sqrt{5}}n\varphi^{2n}+\frac{1}{25}\varphi^{4n}}}, c n ≥ 25 1 φ − 4 n + 25 4 φ − 2 n − 5 5 2 n φ − 2 n − 5 2 + n + 25 4 φ 2 n + 5 5 2 n φ 2 n + 25 1 φ 4 n 1 , φ \varphi φ n → ∞ n\to\infty n → ∞ c n ∼ 5 φ − 2 n c_n\sim 5\varphi^{-2n} c n ∼ 5 φ − 2 n C n C_n C n C n = 1 4 csc 2 ( π 4 n + 2 ) = 4 n 2 π 2 + 4 n π 2 + ( 1 12 + 1 π 2 ) + O ( 1 n 2 ) , n ∈ Z + . C_n=\frac14 \csc^2\bigg(\frac{\pi}{4n+2}\bigg)=\frac{4n^2}{\pi^2}+\frac{4n}{\pi^2}+\bigg(\frac{1}{12}+\frac{1}{\pi^2}\bigg)+\mathcal{O}\bigg(\frac{1}{n^2}\bigg),\quad n\in\mathbb{Z}_+. C n = 4 1 csc 2 ( 4 n + 2 π ) = π 2 4 n 2 + π 2 4 n + ( 12 1 + π 2 1 ) + O ( n 2 1 ) , n ∈ Z + .
引用
@article{arxiv.2002.03337,
title = {Bounds on the spectrum of nonsingular triangular $(0,1)$-matrices},
author = {Vesa Kaarnioja},
journal= {arXiv preprint arXiv:2002.03337},
year = {2020}
}
备注
12 pages, 2 figures