A note on commuting additive maps on rank k symmetric matrices

Abstract

Let $n\geq 2$ and $1<k\leq n$ be integers. Let $S_n(\mathbb{F})$ be the linear space of $n\times n$ symmetric matrices over a field $\mathbb{F}$ of characteristic not two. In this note, we prove that an additive map $\psi:S_n(\mathbb{F})\rightarrow S_n(\mathbb{F})$ satisfies $\psi(A)A=A\psi(A)$ for all rank $k$ matrices $A\in S_n(\mathbb{F})$ if and only if there exists a scalar $\lambda\in \mathbb{F}$ and an additive map $\mu:S_n(\mathbb{F})\rightarrow \mathbb{F}$ such that
\[
\psi(A)=\lambda A+\mu(A)I_n,
\]
for all $A\in S_n(\mathbb{F})$, where $I_n$ is the identity matrix. Examples showing the indispensability of assumptions on the integer $k>1$ and the underlying field $\mathbb{F}$ of characteristic not two are included.