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

Abstract

Let D be any division ring with an involution, ℋn(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 ℋn(D)(n ≥ 2) to itself such that ϕ preserves the adjacency, then ϕ−1 also preserves the adjacency. Moreover, if ℋn(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.