A proof of a conjecture on monotonic behavior of the smallest and the largest eigenvalues of a number theoretic matrix


ALTINIŞIK E. , BÜYÜKKÖSE Ş.

LINEAR ALGEBRA AND ITS APPLICATIONS, vol.471, pp.141-149, 2015 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 471
  • Publication Date: 2015
  • Doi Number: 10.1016/j.laa.2014.12.020
  • Title of Journal : LINEAR ALGEBRA AND ITS APPLICATIONS
  • Page Numbers: pp.141-149

Abstract

In this study we investigate the monotonic behavior of the smallest eigenvalue t(n) and the largest eigenvalue T-n of the n x n matrix E-n(T) E-n, where the ij-entry of E-n is 1 if j vertical bar i and 0 otherwise. We present a proof of the Mattila-Haukkanen conjecture which states that for every n is an element of Z(+), t(n+1) <= t(n) and T-n <= Tn+1. Also, we prove that lim(n ->infinity) t(n) = 0 and lim(n ->infinity) T-n = infinity and we give a lower bound for t(n). (C) 2014 Elsevier Inc. All rights reserved.