Induced Maps on Matrices over Fields

Abstract

Suppose that $𝔽$ is a field and $m,n\ge \mathrm{3}$ are integers. Denote by $M_{mn}(𝔽)$ the set of all $m\times n$ matrices over $𝔽$ and by $M_n(𝔽)$ the set $M_{nn}(𝔽)$. Let $f_{ij}$ ($i\in [\mathrm{1},m],j\in [\mathrm{1},n]$) be functions on $𝔽$, where $[\mathrm{1},n]$ stands for the set $\{\mathrm{1},\dots ,n\}$. We say that a map $f:M_{mn}(𝔽)\to M_{mn}(𝔽)$ is induced by $\{f_{ij}\}$ if $f$ is defined by $f:[a_{ij}]\mapsto [f_{ij}(a_{ij})]$. We say that a map $f$ on $M_n(𝔽)$ preserves similarity if $A~B\Rightarrow f(A)~f(B)$, where $A~B$ represents that $A$ and $B$ are similar. A map $f$ on $M_n(𝔽)$ preserving inverses of matrices means $f(A)f(A^{-\mathrm{1}})=I_n$ for every invertible $A\in M_n(𝔽)$. In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.