Geometry of n × n(n ≥ 3) Hermitian Matrices Over Any Division Ring with an Involution and Its Applications

Abstract

Let D be a division ring with an involution − and char(D) ≠ 2, F = {x ∈ D: x = }. Let ℋ n (D) be the set of all n × n hermitian matrices over D. Two hermitian matrices H 1 and H 2 are said to be “adjacent” if rank(H 1 − H 2) = 1. The fundamental theorem of geometry of hermitian matrices over D is proved: If n ≥ 3 and 𝒜 is a bijective map from ℋ n (D) to itself such that 𝒜 preserves the adjacency, then ∀ X ∈ ℋ n (D), where a ∈ F*, P ∈ GL n (D), and ρ is an automorphism of D which satisfies for all x ∈ D. The application of the fundamental theorem to algebra and geometry is discussed. For example, every Jordan isomorphism or additive rank-1-preserving surjective map on ℋ n (D) is characterized.