Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains

作者：Tan Qianrong^*

来源：Linear and Multilinear Algebra, 2010, 58(5): 659-671.

DOI：10.1080/03081080903008579

• divisibility
• coprime divisor chains
• greatest-type divisor
• power GCD matrix
• power LCM matrix

Let a, b and h be positive integers and S {x1, x2, ... , x(h)} be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation sigma of {1, ... , h} such that x(sigma(1)) vertical bar...vertical bar x(sigma(h)). We say that the set S consists of two coprime divisor chains if we can partition S as S = S(1) boolean OR S(2), where S(1) and S(2) are divisor chains and each element of S(1) is coprime to each element of S(2). The matrix having the a-th power (x(i), x(j))(a) of the GCD (GCD: greatest common divisor) of x(i) and x(j) as its i, j-entry is called a-th power GCD matrix defined on S, denoted by (S(a)). Similarly we can define the a-th power least common multiple (LCM) matrix [S(a)]. In this article, we show the following results: assume that S consists of two coprime divisor chains and 1 is an element of S. We first show that if a vertical bar b, then the power GCD matrix (S(a)) divides the power GCD matrix (S(b)) in the ring M(h)(Z) of h x h matrices over integers. But such factorization should not hold if a (sic) b. Consequently, we show that if a vertical bar b, the power LCM matrix [S(a)] divides the power LCM matrix [S(b)] in the ring M(h)(Z). Finally we show that if a vertical bar b, the power GCD matrix (S(a)) divides the power LCM matrix [S(b)] in the ring M(h)(Z). But such results fail to be true if a(sic)b. These results confirm partially Hong's conjectures.

• 出版日期2010
• 单位攀枝花学院