Induced Maps on Matrices over Fields

Li Yang, Xuezhi Ben, Ming Zhang and Chongguang Cao

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

Abstract: Suppose that 𝔽 is a field and m, n ≥ 3 are integers. Denote by Mmn(𝔽) the set of all m × n matrices over 𝔽 and by Mn(𝔽) the set Mnn(𝔽). Let fij (i ∈ [1, m], j ∈ [1, n]) be functions on 𝔽, where [1, n] stands for the set {1, …, n}. We say that a map f : Mmn(𝔽) → Mmn(𝔽) is induced by {fij} if f is defined by f : [aij] ↦ [fij(aij)]. We say that a map f on Mn(𝔽) preserves similarity if A ~ B⇒f(A) ~ f(B), where A ~ B represents that A and B are similar. A map f on Mn(𝔽) preserving inverses of matrices means f(A)f(A−1) = In for every invertible A ∈ Mn(𝔽). In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.

Date: 2014
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