Divisibility properties of power GCD matrices and power LCM matrices

作者：Hong Shaofang^*

来源：Linear Algebra and Its Applications, 2008, 428(4): 1001-1008.

DOI：10.1016/j.laa.2007.08.037

• divisibility
• divisor chain
• power GCD matrix
• power LCM matrix

Let a, b and n be positive integers and the set S = {x(1), ... , x(n)} of n distinct positive integers be a divisor chain (i.e. there exists a permutation a on {1,..., n} such that x(sigma(1)) vertical bar ... vertical bar x(sigma(n))). In this paper, we show that if a vertical bar b, then the ctth power GCD matrix (S-a) having the ath power (x(i), x(j))a of the greatest common divisor of x(i) and x(j) as its i, j-entry divides the bth power GCD matrix (S-b) in the ring M-n (Z) of n x n matrices over integers. We show also that if a inverted iota b and n > 2, then the ath power GCD matrix (S-a) does not divide the bth power GCD matrix (S-b) in the ring M-n (Z). Similar results are also established for the power LCM matrices.

• 出版日期2008-2-1
• 单位四川大学