Involutary expressions for elements in \(\text{GL}_ n(\mathbb{Z})\) and \(\text{SL}_ n(\mathbb{Z})\)
type: Item
instance of: scholarly article
title: Involutary expressions for elements in \(\text{GL}_ n(\mathbb{Z})\) and \(\text{SL}_ n(\mathbb{Z})\)
zbMATH Open document ID: 0823.20048
DOI: 10.1016/0024-3795(93)00206-F
author: Hiroyuki Ishibashi
published in: Linear Algebra and its Applications
publication date: 1995-05-23T00:00:00Z
review text: Let \(\mathbb{Z}\) be the ring of integers. Consider all matrices with entries in \(\mathbb{Z}\). Let \(\text{GL}_ n (\mathbb{Z})\) and \(\text{SL}_ n (\mathbb{Z})\) be the sets of matrices with determinants \(\pm 1\) and 1, respectively. The author shows that every matrix in \(\text{GL}_ n (\mathbb{Z})\) is a product of \(3n+9\) or fewer involutions and that every matrix in \(\text{SL}_ n (\mathbb{Z})\) is a product of \(3n + 11\) or fewer involutions in \(\text{SL}_ n (\mathbb{Z})\). The proof uses imaginative matrix calculations and an application of Dirichlet's theorem.
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 757614
zbMATH Keywords: integer matrices
zbMATH Keywords: involutions
zbMATH Keywords: factorizations
zbMATH Keywords: general linear groups
zbMATH Keywords: special linear groups
zbMATH Keywords: products of involutions
MaRDI profile type: Publication
cites work: Elementary expressions for unimodular matrices
cites work: Q3228640
cites work: Bireflectionality of the orthogonal group over a valuation domain
cites work: Products of involutions
cites work: Q3993275
description: scientific article; zbMATH DE number 757614