Induced Maps on Matrices over Fields - Li Yang; Xuezhi Ben; Ming Zhang; Chongguang Cao (Abstract and Applied Analysis)
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.