A Note on k-Potence Preservers on Matrix Spaces over Complex Field. (5th June 2013)

• Record Type: Journal Article
• Authors: Song, Xiaofei; Cao, Chongguang; Zheng, Baodong
• Other Names: Zhang Chunrui Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2013(2013)
• Publication Date: 2013-06-05
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2013/581683
• Languages: English
• ISSNs: 1085-3375

Abstract

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