On a generalization of the reciprocal LCM matrix

Abstract

Let S = ${x_1,x_2,...,x_n}$ be a set of distinct positive integers. The nxn matrix 1/[S] = $(s_{ij})$, where $s_{ij} = 1/[x_i,x_j]$, the reciprocal of the least common multiple of $x_i$, and $x_j$, is called the reciprocal least common multiple (reciprocal LCM) matrix on S . In this paper, we present a generalization of the reciprocal LCM matrix on S , that is the matrix $1/[S^r]$, the ij-entry of which is $1/[x_i,x_j]^r$, where r is a real number. We obtain a structure theorem for $1/[S^r]$ and the value of the determinant of $1/[S^r]$. We also prove that $1/[S^r]$ is positive definite if r>0. Then we calculate the inverse of $1/[S^r]$ on a factor closed set. Finally, we show that the matrix $[S^r]=([x_i,x_j]^r)$ defined on S is the product of an integral matrix and the generalized reciprocal LCM matrix $1/[S^r]=(1/[x_i,x_j]^r)$ if S is factor closed and r is a positive integer.