On maximal nilspaces of matrices

Abstract

In 1958, Gerstenhaber showed that if is a subspace of the vector space of the square matrices of order n over some field consisting of nilpotent matrices only (to be called a nilspace) and if the underlying field is sufficiently large, then the maximal dimension of is . This dimension is attained if and only if the linear space is similar to the space of all strictly upper-triangular matrices. In this paper, we study maximal spaces of nilpotent square matrices of order n. As a striking extension of the Gerstenhaber’s result, we prove that a maximal nilspace (with the underlying field being sufficiently large) is similar to a (subspace of) all strictly upper-triangular matrices if and only if it contains a nilpotent J of maximal possible rank and its square . We give a twisted but elementary proof of this fact.