Inverse-preserving Linear Maps Between Spaces of Matrices over Fields.

Suppose F is a field different from F2, the field with two elements. Let Mn (F) and Sn (F) be the space of n × n full matrices and the space of n × n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn (F), Mn (F)} , we say that a linear map f from G1 to G2 is inverse-preserving if f(X)-1 = f(X-1) for every invertible X ∈ G1. Let ... (G1, G2) denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets ... (Sn (F), Mn (F)), ... (Sn (F), Sn (F)), ... (Mn (F), Mn(F)) and are characterized. [ABSTRACT FROM AUTHOR]