The Normal Form of Smith's Matrices

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 $\mathrm{\mid }T\mathrm{\mid }$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n\}$ 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 $((x_i,x_j))\mathrm{\mid }([x_i,x_j])$) 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 $\mathcal{G}$.