On similarity invariants of matrix commutators

Abstract

We study 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. Then we generalize these results in the following way. Let g(X 1,…, X k) be any expression obtained from distinct noncommuting variables X 1,…,X k by applying recursively the Lie product [· ,·] and without using the same variable twice. We study 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.