Upper bounds for the condition numbers of the GCD and the reciprocal GCD matrices in spectral norm

Ipek A.

Computers and Mathematics with Applications, vol.63, no.3, pp.645-651, 2012 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 63 Issue: 3
  • Publication Date: 2012
  • Doi Number: 10.1016/j.camwa.2011.11.016
  • Journal Name: Computers and Mathematics with Applications
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.645-651
  • Keywords: Euler's φ function, GCD matrices, Matrix norms
  • Hatay Mustafa Kemal University Affiliated: Yes

Abstract

Let S= x1,⋯, xn be a set of n distinct positive integers. The n×n matrix having the greatest common divisor ( xi, xj) of xi and xj as its i,j-entry is called the greatest common divisor (GCD) matrix defined on S, denoted by (( xi, xj)), or abbreviated as (S). The n×n matrix (S- 1)=( gij), where gij=1( xi, xj), is called the reciprocal greatest common divisor (GCD) matrix on S. In this paper, we present upper bounds for the spectral condition numbers of the reciprocal GCD matrix (S- 1) and the GCD matrix (S) defined on S=1,2,⋯,n, with n<2, as a function of Euler's φ function and n. © 2011 Elsevier Ltd. All rights reserved.