Non-divisibility of LCM matrices by GCD matrices on gcd-closed sets


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ALTINIŞIK E., Yildiz M., Keskin A.

LINEAR ALGEBRA AND ITS APPLICATIONS, vol.516, pp.47-68, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 516
  • Publication Date: 2017
  • Doi Number: 10.1016/j.laa.2016.11.028
  • Journal Name: LINEAR ALGEBRA AND ITS APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.47-68
  • Keywords: GCD matrix, LCM matrix, Divisibility, Greatest-type divisor, Divisor chain, Mobius function, POWER GCD, DETERMINANTS, NONSINGULARITY, EIGENVALUES, CONJECTURE
  • Gazi University Affiliated: Yes

Abstract

In this paper, we consider the divisibility problem of LCM matrices by GCD matrices in the ring M-n(Z) proposed by Shaofang Hong in 2002 and in particular a conjecture concerning the divisibility problem raised by Jianrong Zhao in 2014. We present some certain gcd-closed sets on which the LCM matrix is not divisible by the GCD matrix in the ring M-n(Z). This could be the first theoretical evidence that Zhao's conjecture might be true. Furthermore, we give the necessary and sufficient conditions on the gcd-closed set S with vertical bar S vertical bar <= 8 such that the GCD matrix divides the LCM matrix in the ring. M-n(Z) and hence we partially solve Hong's problem. Finally, we conclude with a new conjecture that can be thought as a generalization of Zhao's conjecture. (C) 2016 Elsevier Inc. All rights reserved.