# ON THE DIMENSION OF LINEAR SPACES OF NILPOTENT MATRICES

MacDougall, James A. and MacDonald, G. W. and Sweet, L. G. (2010) ON THE DIMENSION OF LINEAR SPACES OF NILPOTENT MATRICES. .

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We obtain bounds on the dimension of a linear space S of nilpotent $n \times n$ 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 and give a new proof of a result Mathes, Omladi�c, 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.