On the smallest singular value in the class of unit lower triangular matrices with entries in [−a, a]
Ercan Altınışık · Special Matrices · 2021
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 c n ( 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?