On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices

ALTINIŞIK E., Keskin A., Yildiz M., Demirbuken M.

LINEAR ALGEBRA AND ITS APPLICATIONS, cilt.493, ss.1-13, 2016 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 493
  • Basım Tarihi: 2016
  • Doi Numarası: 10.1016/j.laa.2015.11.023
  • Dergi Adı: LINEAR ALGEBRA AND ITS APPLICATIONS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.1-13
  • Anahtar Kelimeler: GCD matrix, (0,1)-matrix, Positive matrix, Eigenvalue, Spectral radius, Fibonacci number, ASYMPTOTIC-BEHAVIOR, PROOF
  • Gazi Üniversitesi Adresli: Evet

Özet

Let K-n, be the set of all n x n lower triangular (0, 1)-matrices with each diagonal element equal to 1, L-n = {YYT : Y is an element of K-n} and let c(n) be the minimum of the smallest eigenvalue of YYT as Y goes through K-n. The Ilmonen-Haukkanen-Merikoski conjecture (the IHM conjecture) states that c(n), is equal to the smallest eigenvalue of Y0Y0T, where Y-0 is an element of K-n with (Y-0)(ij) = 1-(-1)(1+j)/2 for i > j. In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices. (C) 2015 Elsevier Inc. All rights reserved.