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

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 $$c_n=\min \{\lambda \mid \lambda \text{ is an eigenvalue of }X{X}^{\mathrm{T}},X\in K_n\},n\in {\mathbb{Z}}_{+}\mathrm{.}$$ 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 $c_n=\mathrm{\parallel }{Z}_n{\mathrm{\parallel }}_2^{-1}$, it is shown that the second largest eigenvalue of $Z_n$ is bounded from above by $\frac{4}{5}$ 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.