On Gerstenhaber's theorem for spaces of nilpotent matrices over a skew field

Abstract

Let K be a skew field, and K_0 be a subfield of the central subfield of K such that K has finite dimension q over K_0. Let V be a K_0-linear subspace of n by n nilpotent matrices with entries in K. We show that the dimension of V is bounded above by q n(n-1)/2, and that equality occurs if and only if V is similar to the space of all n by n strictly upper-triangular matrices over K. This generalizes famous theorems of Gerstenhaber and Serezhkin, which cover the special case K=K_0.

17 pages (final version as will appear in Linear Algebra and its Applications)

References in corpus (1)

• On the matrices of given rank in a large subspace

Cited by in corpus (3)

• Spaces of matrices with few eigenvalues
• From primitive spaces of bounded rank matrices to a generalized Gerstenhaber theorem
• The length and depth of associative algebras