On the Hadamard products of GCD and LCM matrices

Authors : Ahmet Ali ÖÇAL

Pages : 0-0

Doi:10.1501/Commua1_0000000348

View : 15 | Download : 8

Publication Date : 2003-01-01

Article Type : Research Paper

Abstract :Let S = {xı, X2,..., Xn} be a set of distinct positive integers. The matrix insert ignore into journalissuearticles values(S); having the greatest common divisor insert ignore into journalissuearticles values(xi , Xj); of Xi and Xj as its i, j-entry is called the greatest common divisor insert ignore into journalissuearticles values(GCD); matrix on S. The matrix [S] having the least common multiple [xi, Xj] of Xi and Xi as its i, j- entry is called the least common multiple insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(S);oinsert ignore into journalissuearticles values(S);’‘ and [S]o[S] ` n matrices are doubly stochastic matrices and trinsert ignore into journalissuearticles values(insert ignore into journalissuearticles values(S);oinsert ignore into journalissuearticles values(S);-`);=lıinsert ignore into journalissuearticles values(insert ignore into journalissuearticles values(S););= £ Xi-
Keywords : Hadamard products, GCD, LCM matrices