两个拟互素因子链上倒数幂GCD与倒数幂LCM矩阵的非奇异性

Nonsingularity of the reciprocal power GCD matrices and the reciprocal power LCM matrices on two quasi-coprime divisor chains

设S={x_1,x_2,…,x_n}是一个正整数的集合,a是一个正实数.如果一个n阶矩阵的第i行第j列的元素定义为1/(x_i,x_j)~a,其中(x_i,x_j)~a表示S中的元素x_1与x_j的最大公因数的a次幂,则称这个矩阵是定义在S上的倒数幂GCD矩阵,用(1/S^a)表示.类似可定义倒数幂LCM矩阵[1/S^a].作者得到了定义在两个拟互素因子链上的倒数幂GCD矩阵与倒数幂LCM矩阵的行列式公式,并由此证明了定义在两个拟互素因子链上的倒数幂GCD矩阵与倒数幂LCM矩阵均是非奇异的.

Let S = {x1 ,x2,… ,xn } be a set of n distinct positive integers and a ≥ 0 be a real number. The matrix having the ath power 1/（xi,xj）a as its （i,j） -entry is called reciprocal power greatest common divisor （GCD） matrix defined on S, denoted by （1/Sa）. where （xi ,xj ）4 = （gcd（xi ,xj ） ）a. Similarly we can define reciprocal power LCM matrix [1/Sa] on S. In this paper, the authors first obtain formulaes for the determinants of reciprocal power GCD matrix and reciprocal power LCM matrix defined on S which con- sists of two quasi-coprime divisor chains and gcd（S） E S. Then they show that the reciprocal power GCD matrix and the reciprocal power LCM matrix defined on two quasi-coprime divisor chains S with gcd（S） ∈ S are nonsingular.