Representation of a zero trace matrix as a commutator
type: Item
instance of: scholarly article
title: Representation of a zero trace matrix as a commutator
author: Ry Cyna
author: Irwin S. Pressman
published in: Comptes Rendus Mathématiques de l'Académie des Sciences
publication date: 2024-09-10T00:00:00Z
review text: It is clear that the commutator \([A,B] =AB-BA\) of two square matrices \(A,B\) has trace equal to zero. The converse is also true over fields of characteristic zero. Specifically, for any square matrix \(M\) with trace zero, there exist matrices \(H,X\) such that \(M=[H,X]\). This paper describes how to construct these matrices \(H,X\). It is sufficient to find them when \(M\) is a Jordan normal form \(J\). In this case, \(H\) can be given as a matrix with \(1\)'s on the superdiagonal and \(0\) elsewhere. The entries of \(X\) are zero except for the diagonal and sub-diagonal entries. The surviving entries of \(X\) can be described in a simple way from the linear information in \(J\).
reviewed by: Sho Matsumoto
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A16
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 15B05
zbMATH DE Number: 7909468
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: zero trace matrix
zbMATH Keywords: commutator
zbMATH Keywords: matrix exponential
MaRDI profile type: Publication
cites work: Einige Sätze über Matrizen.
cites work: On matrices of trace zero
cites work: A Note on Matrices with Zero Trace
cites work: Matrix Commutators Over an Algebraically Closed Field
cites work: Every matrix is a product of Toeplitz matrices
description: scientific article; zbMATH DE number 7909468