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Let  K  be an arbitrary field and  X  a square matrix over  K . Then  X  is sum of two square nilpotent matrices over  K  if and only if, for every algebraic extension  L  of  K  and arbitrary nonzero  α  ∈  L , there exist idempotent matrices  P   1  and  P   2  over  L  such that  X  =  αP   1  −  αP   2 .

A Note on Decomposing a Square Matrix as Sum of Two Square Nilpotent Matrices over an Arbitrary Field