发表机构
Mathematical College, Sichuan University（四川大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Smith矩阵（GCD矩阵）的对角化，提出定理4.1和4.2，将整除性结果推广到满足条件$\mathcal{G}$的gcd封闭集。
AI 中文摘要
对于任意整数$x$和$y$，设$(x,y)$和$[x,y]$分别表示$x$和$y$的最大公因数和最小公倍数。我们用$|T|$表示有限集合$T$的元素个数。设$a,b$和$n$为正整数，设$S=\{x_1,...,x_n\}$为$n$个不同正整数的集合。设$(f((x_i,x_j)))$（简记为$f(S)$）和$(f([x_i,x_j]))$（简记为$(f([S]))$）分别表示$(i,j)$元为$(f((x_i,x_j)))$和$(f([x_i,x_j]))$的$n\times n$矩阵。1989年，Beslin和Ligh给出了$((x_i,x_j))$的下三角分解的描述。1992年，Bourque和Ligh证明了如果$S$是因子封闭的（即$S$包含$S$中任何元素的所有正除数），则GCD矩阵$((x_i,x_j))$在整数上的$n\times n$矩阵环$M_n(\mathbb{Z})$中整除LCM矩阵$([x_i,x_j])$（记为$((x_i,x_j))|([x_i,x_j])$）。在本文中，我们将展示$((x_i,x_j))$的对角化及其应用。我们的主要新贡献是定理4.1和4.2，它们将先前的结果推广到满足条件$\mathcal{G}$的gcd封闭集。
英文摘要
For any integers $x$ and $y$, let $(x,y)$ and $[x,y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$, respectively. We denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1,...,x_n\}$ be a set of $n$ distinct positive integers. Let $(f((x_i,x_j)))$ (abbreviated by $f(S)$) and $(f([x_i,x_j]))$ (abbreviated by $(f([S]))$) stand for the $n\times n$ matrices whose $(i,j)-$entry is $(f((x_i,x_j)))$ and $(f([x_i,x_j]))$ respectively. In 1989, Beslin and Ligh gave a description of the lower triangular decomposition of $((x_i,x_j))$. In 1992, Bourque and Ligh showed that if $S$ is factor closed (i.e., S contains all positive divisors of any element of S), then the GCD matrix $((x_i,x_j))$ divides the LCM matrix $([x_i,x_j])$ (written as $((x_i,x_j))|([x_i,x_j])$) in the ring $M_n(\mathbb{Z})$ of $n\times n$ matrices over the integers. In this paper, we will show the diagonalization of $((x_i,x_j))$ and its applications. Our main new contributions are Theorems 4.1 and 4.2, which extend previous results to gcd-closed sets satisfying condition $\mathcal{G}$.
