Notes on the divisibility of GCD and LCM Matrices

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.