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

Abstract

Let Kndenote the set of all nonsingular n×nlower triangular (0,1)-matrices. Hong and Loewy (2004) introduced the number sequence cn=min{λ|λ is an eigenvalue ofXXT, X∈Kn}, n∈Z+. There have been a number of attempts in the literature to obtain bounds on the numbers cnby Mattila (2015), Altınışık et al. (2016), Kaarnioja (2021), Loewy (2021), and Altınışık (2021). In this paper, improved upper and lower bounds are derived for the numbers cn. By considering the characteristic polynomial corresponding to the matrix Znsatisfying cn=∥Zn∥−12, it is shown that the second largest eigenvalue of Znis bounded from above by 45leading to an improved upper bound on cn. On the other hand, Samuelson’s inequality applied to the roots of the characteristic polynomial of Znyields an improved lower bound. Numerical experiments demonstrate the quality of the new bounds.