Equivalent Conditions In Theorem Of Geometry Of Hermitian Matrices

The study of geometry of matrices was initiated by Hua L.-K. in forties of the 20thcentury. Scholars such as Wan Z.-X. and Huang L.-P. have proved the fundamentaltheorem of geometry of symmetric matrices over any field and the fundamental theoremof geometry of n×n(n≥2) Hermitian matrices over a division ring of characteristicnot two. Recently, Huang L.-P. has discussed equal conditions on the fundamentaltheorems by graph theory, he defined"good distance graph"and proved the set ofHermitian (resp. symmetric) matrices over a division ring with an involution (resp.field) of characteristic not two is a good distance graph.Based on their work, the geometry of Hermitian matrices over a division ring ofcharacteristic two is discussed in this paper. Let D be a division ring with an involution , ZD be the central field of D, F = {a∈D : a = a}. Denote by H_n(D) a set ofn×n(n≥2) Hermitian matrices over D, by S₃(F₂) a set of 3×3 symmetric matricesover the finite field F₂. Define A～B rank(A B) = 1 A,B∈H_n(D). Then wehave a connected graph (H_n(D),～).This paper has three parts. In chapter 1, we introduce the background, status ofrecent research and main results. In chapter 2, we give a counterexample which is aninverse preserving bijection but it is not an adjacency preserving map in both directions.Chapter 3 discusses the bounded distance preserving maps in both directions on H_n(D)when D satisfying D= F, F ZD and char(D) = 2, proves that the graph (H_n(D),～)is a good distance graph when |F| > 2, and (H_n(F₄),～) is not a good distance graphwhere F₄ is the finite field of 4 elements. A result is given in the last section as following:Let D be a division ring with an involution such that D is not a field of characteristic2 with D = F, and either |D|≥5 or |F∩ZD|≥4. Letφ: H_n(D)â†'H_m(D) be amap. Thenφis an arithmetic distance preserving map if and only ifφis an adjacencypreserving map and there exist P,Q∈H_n(D) such that ad(φ(P),φ(Q)) = n.