The maximum dimension of a subspace of nilpotent matrices of index 2 (Q836994)

An \(n\times n\) matrix \(M\) is nilpotent of index 2 if \(M^2= 0\). The paper deals with linear spaces \(V\) of nilpotent \(n\times n\) matrices of index 2 over a field \(k\) where \(\text{card\,}k> n\). The main result of this paper shows that the maximum dimension of such a space \(V\) depends on the maximum \(r\) of the ranks of matrices in the space and we have \(\dim V\leq r(n-r)\). This bound is sharp and the paper characterizes those spaces attaining this maximum dimension. The authors show that the main result has some important consequences. It gives an immediate proof that \(r(n-r)\) is the maximum possible dimension of a space of anti-commuting matrices over any field \(k\) of \(\text{card\,}k> n/2\) (and \(\text{char\,}k\neq 2\)). It also shows that \(r(n-r)\) is the maximum dimension of a zero subalgebra of a special Jordan algebra. All of the proofs involve only elementary linear algebra.