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

Abstract EN

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.