Abstract
Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽 ≠ 𝔽3. Let M n (𝔽) be the space of all n × n full matrices over 𝔽 and P n (𝔽) the subset of M n (𝔽) consisting of all n × n idempotent matrices and GL n (𝔽) the subset of M n (𝔽) consisting of all n × n invertible matrices. Let Φ𝔽(n, m) denote the set of all maps from M n (𝔽) to M m (𝔽) satisfying A − λB ∈ P n (𝔽) ⇒ φ(A) − λφ(B) ∈ P m (𝔽) for every A, B ∈ M n (𝔽) and λ ∈ 𝔽, where m and n are integers with 3 ≤ n ≤ m. It is shown that if φ ∈ Φ𝔽(n, m), then there exists T ∈ GL m (𝔽) such that φ(A) = T [A ⊗ I p ⊕ A t ⊗ I q ⊕ 0]T −1 for every A ∈ M n (𝔽), where I 0 = 0. This improves the results of some related references.
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Acknowledgements
This work is supported by the Postdoctoral Foundation of China (No. 520-415099) and the NSF of P.R. China (No.10871056). The authors would like to thank the referee for his valuable comments and suggestions to the earlier version of this article.