Bounds on the spectrum of nonsingular triangular (0,1)-matrices

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:178

Article

Kaarnioja, V^1,2 

[1] Univ New South Wales, Sch Math & Stat, Sydney, NSW 2052, Australia

[2] LUT Univ, Sch Engn Sci, POB 20, FI-53851 Lappeenranta, Finland

关键词: Binary matrix;    Singular value;    Semilattice;   

DOI  :  10.1016/j.jcta.2020.105353

来源: Elsevier

【 摘 要 】

Let K, be the set of all nonsingular n x n lower triangular (0, 1)-matrices. Hong and Loewy (2004) introduced the numbers c(n) = min{lambda vertical bar lambda is an eigenvalue of XXT, X is an element of K-n}, n is an element of Z(+). A related family of numbers was considered by Ilmonen, Haukkanen, and Merikoski (2008): C-n = max{lambda vertical bar lambda is an eigenvalue of XXT, X is an element of K-n}, n is an element of 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 cn >= 1/root 1/25 phi(-4n) + 2/25 phi(-2n) - 2/5 root 5 n phi(-2n) - 23/25 + n + 2/25 phi(2n )+ 2/5 root 5 n phi(2n )+( )1/25 phi(4n,) and for n even, one has cn >= 1/root 1/25 phi(-4n) + 4/25 phi(-2n) - 2/5 root 5 n phi(-2n) - 2/5 + n + 4/25 phi(2n )+ 2/5 root 5 n phi(2n )+( )1/25 phi(4n,) where phi denotes the golden ratio. These lower bounds improve the estimates derived previously by Mattila (2015) and Altinisik et al. (2016). The sharpness of these lower bounds is assessed numerically and it is conjectured that c(n) similar to 5 phi(-2n )as n -> infinity. In addition, a new closed form expression is derived for the numbers C-n, viz. C-n = 1/4 csc(2) (pi/4n + 2) = 4n(2)/pi(2) + 4n/pi(2) + (1/12 + 1/pi(2)) + O(1/n(2)), n is an element of Z(+). (C) 2020 Elsevier Inc. All rights reserved.

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