Induced Maps on Matrices over Fields

Suppose that F is a field and m , n ≥ 3 are integers. Denote by M m n ( F ) the set of all m × n matrices over F and by M n ( F ) the set M n n ( F ) . Let f i j ( 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 : M m n ( F ) → M m n ( F ) 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 ( F ) 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 ( F ) preserving inverses of matrices means f ( A ) f ( A - 1 ) = I n for every invertible A ∈ M n ( F ) . In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.