ON BOUNDS FOR THE SMALLEST AND THE LARGEST EIGENVALUES OF GCD AND LCM MATRICES


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ALTINIŞIK E. , BÜYÜKKÖSE Ş.

MATHEMATICAL INEQUALITIES & APPLICATIONS, vol.19, no.1, pp.117-125, 2016 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 19 Issue: 1
  • Publication Date: 2016
  • Doi Number: 10.7153/mia-19-09
  • Title of Journal : MATHEMATICAL INEQUALITIES & APPLICATIONS
  • Page Numbers: pp.117-125
  • Keywords: gcd matrix, lcm matrix, Euler's phi-function, eigenvalue inequalities, trace, Cauchy's interlacing theorem, Rayleigh-Ritz theorem, COMMON DIVISOR MATRICES, JOIN MATRICES, ASYMPTOTIC-BEHAVIOR, TRACES, DETERMINANT, CONJECTURE, PROOF

Abstract

In this paper, improving a famous result of Wolkowicz and Styan for the GCD matrix (S-n) and the LCM matrix [S-n] defined on S-n = {1,2,..., n}, we present new upper and lower bounds for the smallest and the largest eigenvalues of (S-n) and [S-n] in terms of particular arithmetical functions.