International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 6, Pages 925-935
doi: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 Pentti Haukkanen and Ismo Korkee. 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(ℤ) of the n×n matrices over the integers.