Induced Maps on Matrices over Fields

Abstract

Suppose that𝔽is a field andm,n≥3are integers. Denote byMmn(𝔽)the set of allm×nmatrices over𝔽and byMn(𝔽)the setMnn(𝔽). Letfij(i∈[1,m],j∈[1,n]) be functions on𝔽, where[1,n]stands for the set{1,…,n}. We say that a mapf:Mmn(𝔽)→Mmn(𝔽)is induced by{fij}iffis defined byf:[aij]↦[fij(aij)]. We say that a mapfonMn(𝔽)preserves similarity ifA~B⇒f(A)~f(B), whereA~Brepresents thatAandBare similar. A mapfonMn(𝔽)preserving inverses of matrices meansf(A)f(A-1)=Infor every invertibleA∈Mn(𝔽). In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.