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, cilt.471, ss.141-149, 2015 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 471
  • Basım Tarihi: 2015
  • Doi Numarası: 10.1016/j.laa.2014.12.020
  • Dergi Adı: LINEAR ALGEBRA AND ITS APPLICATIONS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.141-149
  • Anahtar Kelimeler: GCD matrix, LCM matrix, Eigenvalue, Eigenvalue inequalities, Matrix norm, COMMON DIVISOR MATRICES, JOIN MATRICES, LCM MATRICES, GCD MATRICES
  • Gazi Üniversitesi Adresli: Evet

Özet

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.