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.

Keywords:

• matrix
• subspace
• nilpotent
• maximal nilspace
• rank

AMS Subject Classification:

• Primary: 15A30
• 15B48
• Secondary: 47D03

Acknowledgements

This work was supported by a grant from the Slovenian Research Agency – ARRS. The author is deeply grateful to Janez Bernik, Laurent Marcoux and Mitja Mastnak for many useful discussions on the problem, and especially to Heydar Radjavi, who also had the starting idea of what should be done and the neverending power to make us think about it.