On the dimension of linear spaces of nilpotent matrices

We obtain bounds on the dimension of a linear space S of nilpotent n×n matrices over an arbitrary field. We consider the case where bounds k and r are known for the nilindex and rank, respectively, and find the best possible dimensional bound on the subspace S in terms of the quantities n, k and r. We also consider the case where information is known concerning the Jordan forms of matrices in S and obtain new dimensional bounds in terms of this information. These bounds improve known bounds of Gerstenhaber. Along the way, we generalize a result of Mathes, Omladič, and Radjavi concerning traces on subspaces of nilpotent matrices. This is a key component in the proof of our result and may also be of independent interest.