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The Normal Form of Smith's Matrices

  • Wenzhong Lei
  • Han Zhang
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Abstract

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={x1,...,xn} be a set of n distinct positive integers. Let (f((xi,xj))) (abbreviated by f(S)) and (f([xi,xj])) (abbreviated by (f([S]))) stand for the n× n matrices whose (i,j)−entry is (f((xi,xj))) and (f([xi,xj])) respectively. In 1989, Beslin and Ligh gave a description of the lower triangular decomposition of ((xi,xj)). 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 ((xi,xj)) divides the LCM matrix ([xi,xj]) (written as ((xi,xj))|([xi,xj])) in the ring Mn(ℤ) of n× n matrices over the integers. In this paper, we will show the diagonalization of ((xi,xj)) and its applications. Our main new contributions are Theorems 4.1 and 4.2, which extend previous results to gcd-closed sets satisfying condition 𝒢.

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Cite

BibTeXRIS
BibTeX
@misc{lei2026normal,
  title = {{The Normal Form of Smith's Matrices}},
  author = {Wenzhong Lei and Han Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07352},
  eprint = {2609.07352},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07352},
}
RIS
TY  - UNPB
TI  - The Normal Form of Smith's Matrices
AU  - Wenzhong Lei
AU  - Han Zhang
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07352
UR  - https://arxiv.org/abs/2609.07352
AB  - 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× 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(ℤ) of n× 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 𝒢.
ER  -