Affiliation:
1. Department of Mathematics, Faculty of Sciences , Gazi University 06500 Teknikokullar - Ankara , Turkey
Abstract
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?
Subject
Geometry and Topology,Algebra and Number Theory
Reference12 articles.
1. [1] E. Altınışık, On a conjecture on the smallest eigenvalues of some special positive definite matrices, 3rd International Conference on Applied Mathematics & Approximation Theory - AMAT 2015, 28-31 May 2015, Ankara, Turkey.
2. [2] E. Altınışık and Ş. Büyükköse, A proof of a conjecture on monotonic behavior of the smallest and the largest eigenvalues of a number theoretic matrix, Linear Algebra Appl. 471, 141–149 (2015).
3. [3] E. Altınışık and Ş. Büyükköse, On bounds for the smallest and the largest eigenvalues of GCD and LCM matrices, Math. Inequal. Appl. 19, 117–125 (2016).
4. [4] E. Altınışık, A. Keskin, M. Yıldız and M. Demirbüken, On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices, Linear Algebra Appl. 493, 1–13 (2016).
5. [5] E. Altınışık, A. Keskin and M. Yıldız, A note on the smallest eigenvalues of a certain class of matrices, unpublished preprint, 5 pages (2016).