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

作　　者：Li Ping HUANG 

机构地区：[1]School of Mathematics, Changsha University of Science & Technology Changsha 410076, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2007年第1期95-102,共8页数学学报（英文版）

基　　金：the National Natural Science Foundation of China Grant,#10271021

摘　　要：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） ≠J3（F2）, then φ preserves the arithmetic distance. Thus, an open problem posed by Wan Zhe-Xian is answered for geometry of symmetric and hermitian matrices.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） ≠J3（F2）, then φ preserves the arithmetic distance. Thus, an open problem posed by Wan Zhe-Xian is answered for geometry of symmetric and hermitian matrices.

关 键 词：division ring with involution hermitian inatrix ADJACENCY geometry of matrices 

分 类 号：O151.21[理学—数学]