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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 P1 and P2 over L such that X=αP1−αP2.

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