Induced Maps on Matrices over Fields

Abstract EN

Suppose that ? is a field and m , n ≥ 3 are integers.

Denote by M m n ( ? ) the set of all m × n matrices over ? and by M n ( ? ) the set M n n ( ? ) .

Let f i j ( i ∈ [ 1 , m ] , j ∈ [ 1 , n ] ) be functions on ? , where [ 1 , n ] stands for the set { 1 , … , n } .

We say that a map f : M m n ( ? ) → M m n ( ? ) is induced by { f i j } if f is defined by f : [ a i j ] ↦ [ f i j ( a i j ) ] .

We say that a map f on M n ( ? ) preserves similarity if A ~ B ⇒ f ( A ) ~ f ( B ) , where A ~ B represents that A and B are similar.

A map f on M n ( ? ) preserving inverses of matrices means f ( A ) f ( A - 1 ) = I n for every invertible A ∈ M n ( ? ) .

In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.