On the smallest singular value in the class of unit lower triangular matrices with entries in [−a, a]

Abstract

Given a real number a ≥ 1, let Kn(a) be the set of all n × n unit lower triangular matrices with each element in the interval [−a, a]. Denoting by λn(·) the smallest eigenvalue of a given matrix, let cn(a) = min {λ n(YYT) : Y ∈ Kn(a)}. Then cn(a)\sqrt {{c_n}\left( a \right)} is the smallest singular value in Kn(a). We find all minimizing matrices. Moreover, we study the asymptotic behavior of cn(a) as n → ∞. Finally, replacing [−a, a] with [a, b], a ≤ 0 < b, we present an open question: Can our results be generalized in this extension?

Keywords

• real symmetric matrix
• unit lower triangular matrix
• gcd and lcm matrix
• smallest eigenvalue
• smallest singular value
• 11c20
• 15a18
• 15a42
• 15b36