On a generalization of the reciprocal LCM matrix

Authors : E ALTINIŞIK

Pages : 0-0

Doi:10.1501/Commua1_0000000360

View : 36 | Download : 10

Publication Date : 2002-01-01

Article Type : Research Paper

Abstract :Let S = {x,,x2,...,x,} be a set of distinct positive integers. The matrix 1 /[S] = insert ignore into journalissuearticles values(s .);, where sa = l/[x„X j], the reciprocal of the least common multiple of x, and x ,, is called the reciprocal least common multiple insert ignore into journalissuearticles values(reciprocal LCM); matrix on S . In this paper, we present a generalization of the reciprocal LCM m atrixon S , that is the matrix 1 /[£ `], the ij- entry ofw hichis l/[ x ,,x jr , where r is a real number. We obtain a structure theorem for l/[5 `] and the value of the determinant of 1 /[S` ]. We also prove that 1/[S`] is positive definite if r > 0 . Then we calculate the inverse of l/[5 r ] on a factor closed set. Finally, we show that the matrix [•$`] = insert ignore into journalissuearticles values([ x ,,x j`); defined on S is the product of an integral matrix and the generalized reciprocal LCM matrix l/[5,r ] = insert ignore into journalissuearticles values(l/[x ,,x j`); if S is factor closed and r is a positive integer.
Keywords : The GCD matrix, the LCM matrix, the reciprocal GCD matrix

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