Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains
type: Item
instance of: scholarly article
title: Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains
zbMATH Open document ID: 1221.11002
author: Qianrong Tan
author: Zong Bing Lin
published in: Applied Mathematics and Computation
publication date: 2011-01-14T00:00:00Z
review text: Let \(a,n\geq 1\) be integers, \(S=\{x_1,\dots,x_n\}\) a set of \(n\) distinct positive integers. The matrix having the \(a\)th power \((x_i,x_j)^a\) of the greatest common divisor of \(x_i\) and \(x_j\) as its \(i,j\)-entry is called \(a\)th power greatest common divisor (GCD) matrix defined on \(S\), denoted by \((S^a)\). Similarly one defines the \(a\)th power LCM matrix \( [S^a]\). The authors show that \(\det(S^a)\mid \det(S^b)\), \(\det[S^a]\mid \det[S^b] \) and \(\det(S^a)\mid \det[S^b]\) if \(a\mid b\) and \(S\) consists of finitely many quasi-coprime divisor chains with \(\gcd(S)\in S\).
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11A05
zbMATH DE Number: 5836683
zbMATH Keywords: greatest common divisor
zbMATH Keywords: least common multiple
MaRDI profile type: Publication
full work available at URL: j.amc.2010.09.053
OpenAlex ID: W2079989139
cites work: On GCD and LCM matrices
cites work: Matrices associated with classes of arithmetical functions
cites work: Matrices associated with arithmetical functions
cites work: Matrices associated with multiplicative functions
cites work: On Hong’s conjecture for power LCM matrices
cites work: Q5699234
cites work: Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets
cites work: More on divisibility of determinants of LCM matrices on GCD-closed sets
cites work: Determinants of multiplicative Toeplitz matrices
cites work: On the Bourque-Ligh conjecture of least common multiple matrices
cites work: Gcd-closed sets and determinants of matrices associated with arithmetical functions
cites work: On the factorization of LCM matrices on gcd-closed sets
cites work: Factorization of matrices associated with classes of arithmetical functions
cites work: Divisibility of determinants of least common multiple matrices on GCD-closed sets
cites work: Notes on power LCM matrices
cites work: Nonsingularity of matrices associated with classes of arithmetical functions
cites work: Nonsingularity of least common multiple matrices on gcd-closed sets
cites work: Divisibility properties of power GCD matrices and power LCM matrices
cites work: Infinite divisibility of Smith matrices
cites work: ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF RECIPROCAL POWER LCM MATRICES
cites work: ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF GREATEST COMMON DIVISOR MATRICES
cites work: On Nonsingular Power LCM Matrices
cites work: Divisibility properties of Smith matrices
cites work: Power GCD Matrices for a UFD
cites work: Notes on Hong's conjectures of real number power LCM matrices
cites work: LU-FACTORIZATIONS OF SYMMETRIC MATRICES WITH APPLICATIONS
cites work: Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains
cites work: Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains II
cites work: Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains
cites work: On the divisibility of power LCM matrices by power GCD matrices
DOI: 10.1016/J.AMC.2010.09.053
Recommended article: Determinants and divisibility of power GCD and power LCM matrices on finitely many coprime divisor chains
Recommended article: Q5017592
Recommended article: Divisibility among power GCD matrices and among power LCM matrices on finitely many coprime divisor chains
Recommended article: Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains
Recommended article: Divisibility among power GCD matrices and among power LCM matrices on two coprime divisor chains II
Recommended article: Q3054656
Recommended article: Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains
Recommended article: On the divisibility of power LCM matrices by power GCD matrices
Recommended article: Notes on the non-divisibility of determinants of power GCD and power LCM matrices
Recommended article: Q3110740
description: scientific article; zbMATH DE number 5836683