Integral similarity and commutators of integral matrices
type: Item
instance of: scholarly article
title: Integral similarity and commutators of integral matrices
zbMATH Open document ID: 0801.15015
DOI: 10.1016/0024-3795(94)90509-6
author: Thomas J. Laffey
author: Robert Reams
published in: Linear Algebra and its Applications
publication date: 1994-04-19T00:00:00Z
review text: It is well known and easily proved that the trace function on a matrix algebra over a field is determined, up to scalar multiples, as that unique linear function which annihilates all commutators. In the same context, but less readily proved, is that any traceless matrix can be expressed as a commutator -- Shoda's theorem. It is shown in this paper that the result is also true if the field is replaced by the ring of integers. A related result, in the case where the matrices are over a field, is that a non-scalar traceless matrix is similar to one with zero diagonal. This theorem persists, with minor restrictions, when the field is replaced by the ring of integers. The proofs of these results involve a nice mixture of simple matrix computations, some manipulations of canonical forms, and an application of Dirichlet's theorem on the existence of primes in arithmetic progressions. The subject is not yet exhausted; indeed, the authors speculate on extensions to more general rings.
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 553981
zbMATH Keywords: integer matrix
zbMATH Keywords: integral similarity
zbMATH Keywords: trace function
zbMATH Keywords: matrix algebra
zbMATH Keywords: commutators
zbMATH Keywords: traceless matrix
zbMATH Keywords: canonical forms
reviewed by: N. B. Backhouse
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90509-6
OpenAlex ID: W1997459590
cites work: On matrices of trace zero
cites work: Choices for A in the Matrix Equation T=AB-BA
cites work: Q1252893
cites work: Gauss’ class number problem for imaginary quadratic fields
cites work: Q3892429
cites work: Some general algorithms. I: Arithmetic groups
cites work: A correspondence between classes of ideals and classes of matrices
cites work: Matrices Over Polynomial Rings
cites work: On a Theorem of Latimer and Macduffee
cites work: A rational canonical pair form for a pair of symmetric matrices over an arbitrary field F with char F ≠ 2 and applications to finest simultaneous block diagonalizations
cites work: Q3822313
description: scientific article; zbMATH DE number 553981