Abstract
A matrix M is nilpotent of index 2 if M 2 = 0 . Let V be a space of nilpotent n × n matrices of index 2 over a field k where 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 dim V ⩽ 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.
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