A NOTE ON COMMUTING ADDITIVE MAPS ON RANK K SYMMETRIC MATRICES

Let n≥2 and 1<k≤n be integers. Let S_n(F) be the linear space of n×n symmetric matrices over a field F of characteristic not two. In this note, we prove that an additive map ψ:Sn(F)→S_n(F) satisfies ψ(A)A=Aψ(A) for all rank k matrices A∈Sn(F) if and only if there xists a scalar λ∈F and an additive map μ:S_n(F)→F such thatψ(A)=λA+μ(A)In,for all A∈Sn(F), here In is the identity matrix. Examples showing the indispensability of assumptions on the integer k>1 and the underlying field F of characteristic not two are included.