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

Let ℂ be the field of all complex numbers, Mn the space of all n × n matrices over ℂ, and Sn the subspace of Mn consisting of all symmetric matrices. The map ϕ : Sn → Mn satisfies that A − λB is k‐potent in Sn implying that ϕ(A) − λϕ(B) is k‐potent in Mn, where λ ∈ ℂ, then there exist an invertible matrix P ∈ Mn and ϵ ∈ ℂ with ϵk = ϵ such that ϕ(X) = ϵP−1(X)P for every X ∈ Sn. Moreover, the inductive method used in this paper can be used to characterise similar maps from Mn to Mn.