A Note on k -Potence Preservers on Matrix Spaces over Complex Field

Abstract

Let $ℂ$ be the field of all complex numbers, $M_n$ the space of all $n\times n$ matrices over $ℂ$, and $S_n$ the subspace of $M_n$ consisting of all symmetric matrices. The map $\varphi :S_n\to M_n$ satisfies that $A-\lambda B$ is $k$-potent in $S_n$ implying that $\varphi (A)-\lambda \varphi (B)$ is $k$-potent in $M_n$, where $\lambda \in \mathrm{ℂ,}$ then there exist an invertible matrix $P\in M_n$ and $ϵ\in ℂ$ with ${ϵ}^k=ϵ$ such that $\varphi (X)=ϵP^{-1}(X)P$ for every $X\in S_n$. Moreover, the inductive method used in this paper can be used to characterise similar maps from $M_n$ to $M_n$.