Idempotence-preserving maps without the linearity and surjectivity assumptions

✍ Scribed by Xian Zhang

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 237 KB
• Volume: 387
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2004.02.011

✦ Synopsis

Let M n (F) be the space of all n × n matrices over a field F of characteristic not 2, and let P n (F) be the subset of M n (F) consisting of all n × n idempotent matrices. We denote by n (F) the set of all maps from M n (F) to itself satisfying A -λB ∈ P n (F) if and only if φ(A)λφ(B) ∈ P n (F) for every A, B ∈ M n (F) and λ ∈ F. It was shown that φ ∈ n (F) if and only if there exists an invertible matrix P ∈ M n (F) such that either φ(A) = P AP -1 for every A ∈ M n (F), or φ(A) = P A T P -1 for every A ∈ M n (F). This improved Dolinar's result by omitting the surjectivity assumption and extending the complex field to any field of characteristic not 2.