Sums of two square-zero matrices over an arbitrary field

AbstractThe problem to express an n×n matrix A as the sum of two square-zero matrices was first investigated by Wang and Wu [2] for matrices over the complex field. This paper investigates the problem over an arbitrary field F. It is shown that, if char(F)≠2, then A∈Mn(F) is the sum of two square-zero matrices if and only if A is similar to a matrix of the form N⊕X⊕(-X)⊕⊕i=1mC(gi(x2)), where N is nilpotent, X is nonsingular, and each C(gi(x2)) is a companion matrix associated with an even-power poly nomial with nonzero constant term. If F is of characteristic two, the term X⊕(-X) falls away. If F is of characteristic zero and algebraically closed, the term ⊕i=1mC(gi(x2)) falls away and the result of Wang and Wu is obtained