Powers of 1-cyclic conjugacy classes in the groups GL\(_ n(F)\) and SL\(_ n(F)\)
type: Item
instance of: scholarly article
title: Powers of 1-cyclic conjugacy classes in the groups GL\(_ n(F)\) and SL\(_ n(F)\)
zbMATH Open document ID: 0882.15008
DOI: 10.1016/S0024-3795(96)00206-6
author: Arieh Lev
published in: Linear Algebra and its Applications
publication date: 1998-03-12T00:00:00Z
review text: The author defines an invertible \(n\times n\) matrix \(A\) over a field \(F\) to be 1-cyclic if it is similar to a block-direct sum of cyclic matrices, with at most one of the blocks of size \(1\times 1\). It is shown that if \(A\) is 1-cyclic with \(n\geq 2\) and \(|F|\geq 4\), then every nonscalar \(M\in\text{GL}_n(F)\) with \(\text{det }M=(\text{det }A)^4\) is a product of four matrices, each of which is similar to \(A\) by an element of \(\text{SL}_n(F)\). The result is applied to the problem of factoring elements of \(\text{SL}_n(F)\) into products of unipotent matrices of index two, and into products of matrices of a given finite order.
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 1011328
zbMATH Keywords: 1-cyclic matrix
zbMATH Keywords: determinant
zbMATH Keywords: conjugacy class
zbMATH Keywords: factorization
zbMATH Keywords: products of unipotent matrices
reviewed by: William H. Gustafson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1011328