Representation of a zero trace matrix as a commutator
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 \(J\).
scientific article; zbMATH DE number 7909468