Abstract
Given an n×n real symmetric matrix A, let λn(A) denote its smallest eigenvalue. Let Kn denote the set of all n×n invertible, lower triangular (0,1) matrices, and cn:=min{λn(YYt):Y∈Kn}. Then, cn is the smallest singular value in Kn. Hong and Loewy introduced cn as a mean to obtain inequalities involving eigenvalues of certain GCD (greatest common divisor) and LCM (least common multiple) matrices. Since then, cn has been used in many papers to obtain additional spectral inequalities for GCD and LCM matrices, and their generalizations. Due to its wide spread, it became important to obtain good bounds for cn. In this paper we obtain such bounds, and consequently determine the asymptotic behavior of cn, proving a conjecture of Kaarnioja. Moreover, we prove the uniqueness of the matrix Y∈Kn for which cn is attained, proving a conjecture of Altinisik, Keskin, Yildiz and Demirbüken.
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