Adjacency Preserving Bijection Maps of Hermitian Matrices over any Division Ring with an Involution.

Let D be any division ring with an involution, ℋ n ( D) be the space of all n × n hermitian matrices over D. Two hermitian matrices A and B are said to be adjacent if rank( A − B) = 1. It is proved that if ϕ is a bijective map from ℋ n ( D)( n ≥ 2) to itself such that ϕ preserves the adjacency, then ϕ −1 also preserves the adjacency. Moreover, if ℋ n ( D ≠ $${\fancyscript S}$$ 3( $${\fancyscript F}$$ 2), then ϕ preserves the arithmetic distance. Thus, an open problem posed by Wan Zhe–Xian is answered for geometry of symmetric and hermitian matrices. [ABSTRACT FROM AUTHOR]