On the Hadamard products of GCD and LCM matrices

Abstract

Let S = ${x_1, X_2,..., X_n}$ be a set of distinct positive integers. The matrix (S) having the greatest common divisor $(x_i , X_j)$ of $x_i$ and $x_j$ as its i, j-entry is called the greatest common divisor (GCD) matrix on S. The matrix [S] having the least common multiple $[x_i, x_j]$ of $x_i$ and $x_j$ as its i, j-entry is called the least common multiple (LCM) matrix on S. In this paper we obtain some results related with Hadamard products of GCD and LCM matrices. The set S is factor-closed if it contains every divisor of each of its elements. It is well-known,that if S is factor-closed,then there exit the inverses of the GCD and LCM matrices on S. So we conjecture that if the set S is factor-closed, then $(S)o(S)^{-1}$ and $[S]o[S]^{-1}$ matrices are doubly stochastic matrices and $tr((S)o(S)^{-1} = tr((S)) = sum_{i=1}^n x_i$ .