On Smith's determinant
type: Item
instance of: scholarly article
title: On Smith's determinant
zbMATH Open document ID: 0883.15002
DOI: 10.1016/S0024-3795(96)00192-9
author: Pentti Haukkanen
author: Jun Wang
author: Juha Sillanpää
published in: Linear Algebra and its Applications
publication date: 1998-03-22T00:00:00Z
review text: The authors give a brief review of papers relating to Smith's determinant [cf. \textit{H. J. S. Smith}, Proc. Lond. Math. Soc. 7, 208--212 (1876; JFM 08.0074.02)] and point out a common structure that can be found in many extensions and analogues of Smith's determinant. The common structure is presented in the language of posets. The authors also make an investigation on a conjecture of \textit{S. Beslin} and \textit{S. Ligh} [Bull. Aust. Math. Soc. 40, No. 3, 413--415 (1989; Zbl 0675.10002); Fibonacci Q. 30, No. 2, 157--160 (1992; Zbl 0752.11012)] on greatest common divisor (GCD) matrices in the sense of meet matrices and give characterizations of the posets satisfying the conjecture. Further, a counterexample for the conjecture of \textit{K. Bourque} and \textit{S. Ligh} [Linear Algebra Appl. 174, 65--74 (1992; Zbl 0761.15013)] that the least common multiple matrix on any GCD-closed set is invertible is given.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 11C20
zbMATH DE Number: 1026442
zbMATH Keywords: greatest common divisor matrices
zbMATH Keywords: Smith's determinant
zbMATH Keywords: posets
zbMATH Keywords: meet matrices
zbMATH Keywords: counterexamples
zbMATH Keywords: least common multiple matrix
reviewed by: Rodica Covaci
MaRDI profile type: Publication
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full work available at URL: s0024-3795(96)00192-9
OpenAlex ID: W1969921015
description: scientific article; zbMATH DE number 1026442