The Maximum Dimension of a Subspace of Nilpotent Matrices of Index 2

MacDougall, James A. and Sweet, L. G. (2009) The Maximum Dimension of a Subspace of Nilpotent Matrices of Index 2. Linear Algebra & Applications, 431 (8). pp. 1116-1124.

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A matrix $M$ is nilpotent of index 2 if $M^2 = 0$. Let $V$ be a space of nilpotent $n \times n$ matrices of index 2 over a field $k$ where {emp \card} $k > n$ and suppose that $r$ is the maximum rank of any matrix in $V$ . The object of this paper is to give an elementary proof of the fact that {emp \dim} $V \leq r(n-r)$. We show that the inequality is sharp and construct all such subspaces of maximum dimension. We use the result to find the maximum dimension of spaces of anti-commuting matrices and zero subalgebras of special Jordan Algebras.