New upper and lower bounds on the smallest singular values of nonsingular lower triangular $(0,1)$-matrices

Abstract

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 $$ c_n=\min\{\lambda\mid\lambda~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad n\in\mathbb Z_+. $$ 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.