From primitive spaces of bounded rank matrices to a generalized Gerstenhaber theorem

TL;DR

This paper extends Gerstenhaber's theorem on nilpotent matrices by leveraging Atkinson's structure theorem, broadening its applicability under certain field conditions.

Contribution

It introduces a generalized version of Gerstenhaber's theorem based on primitive spaces of bounded rank matrices, under mild field size assumptions.

Findings

Generalized Gerstenhaber's theorem established

Applicable to broader classes of matrices

Depends on field cardinality conditions

Abstract

A recent generalization of Gerstenhaber's theorem on spaces of nilpotent matrices is derived, under mild conditions on the cardinality of the underlying field, from Atkinson's structure theorem on primitive spaces of bounded rank matrices.