On similarity invariants of matrix commutators and Jordan products

Abstract

Denote by [ X, Y] the additive commutator XY − YX of two square matrices X, Y over a field F. In a previous paper, the possible eigenvalues, ranks and numbers of nonconstant invariant polynomials of [⋯[[ A, X 1], X 2], …, X k ], when A is a fixed matrix and X 1, …, X k vary, were studied. Moreover given any expression g( X 1, …, X k ), obtained from distinct noncommuting variables X 1, …, X k by applying recursively the Lie product [· , ·] and without using the same variable twice, the possible eigenvalues, ranks and numbers of nonconstant invariant polynomials of g( X 1, …, X k ) when one of the variables X 1, …, X k takes a fixed value in F n× n and the others vary, were studied. The purpose of the present paper is to show that analogous results can be obtained when additive commutators are replaced with multiplicative commutators or Jordan products.