GCD封闭集上倒数幂GCD矩阵的非奇异性  

Nonsingularity of Reciprocal Power GCD Matrices on Gcd-Closed Sets

作　　者：朱光艳 ZHU Guangyan(School of Teacher Education,Hubei Minzu University,Enshi 445000,Hubei,China)

机构地区：[1]湖北民族大学教师教育学院,湖北恩施445000

出　　处：《武汉大学学报（理学版）》2022年第5期459-462,共4页Journal of Wuhan University:Natural Science Edition

基　　金：国家自然科学基金(11771304)

摘　　要：设α为正整数,S={x_(1),…,x_(n)}为n个不同的正整数构成的集合。若对任意1≤i,j≤n,均有gcd(x_(i),x_(j))∈S,则称S是GCD封闭集。如果一个n阶方阵的第i行第j列元素为1/(gcd(x_(i),x_(j)))^(α),那么称它是定义在集合S上的α次倒数幂GCD矩阵。本文给出了某些特殊类型的GCD封闭集S上α次倒数幂GCD矩阵的行列式的计算公式,并且得到了有关其非奇异性的一些结果。Letαbe a positive integer and let S={x_(1),…,x_(n)}be a set of n distinct positive integers.The n×n matrix having the reciprocalαth power of the greatest common divisor(GCD)of x_(i)and x_(j)as its(i,j)-entry is called reciprocalαth power GCD matrix defined on S.In this paper,we give the formula for the determinant of reciprocalαth power GCD matrix if S is a gcdclosed set(i.e.gcd(x_(i),x_(j))∈S for all integers i and j with 1≤i,j≤n).Moreover,we establish some results on the nonsingularity of reciprocalαth power GCD matrix defined on certain classes of gcd-closed sets S.

关 键 词：非奇异性 倒数幂GCD矩阵 LCM幂矩阵 GCD封闭集 

分 类 号：O151.21[理学—数学]