Adjacency preserving mappings of symmetric and hermitian matrices

Abstract

Let D be a division ring with an involution \( \overline{\phantom{e}} \) and \( F = \{a \in D \mid \overline{a} = a\} \) . When \( \overline{\phantom{e}} \) is the identity map then D = F is a field and we assume \( \text{\rm char}(F)\neq 2 \) . When \( \overline{\phantom{e}} \) is not the identity map we assume that F is a subfield of D and is contained in the center of D. Let n be an integer, \( n \geq 2 \) , and \( {\mathcal H}_{n}(D) \) be the space of hermitian matrices which includes the space \( {\mathcal S}_{n}(F) \) of symmetric matrices as a particular case. If a bijective map \( \varphi \) of \( {\mathcal H}_{n}(D) \) preserves the adjacency then also \( \varphi^{-1} \) preserves the adjacency.