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International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 6, Pages 925-935
http://dx.doi.org/10.1155/IJMMS.2005.925

Notes on the divisibility of GCD and LCM Matrices

Department of Mathematics, Statistics and Philosophy, University of Tampere, Tampere 33014, Finland

Received 10 November 2004

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let S={x1,x2,,xn} be a set of positive integers, and let f be an arithmetical function. The matrices (S)f=[f(gcd(xi,xj))] and [S]f=[f(lcm[xi,xj])] are referred to as the greatest common divisor (GCD) and the least common multiple (LCM) matrices on S with respect to f, respectively. In this paper, we assume that the elements of the matrices (S)f and [S]f are integers and study the divisibility of GCD and LCM matrices and their unitary analogues in the ring Mn() of the n×n matrices over the integers.