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- Title
Adjacency Preserving Bijection Maps of Hermitian Matrices over any Division Ring with an Involution.
- Authors
Huang, Li Ping
- Abstract
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.
- Subjects
DIVISION rings; RINGS with involution; HERMITIAN operators; SYMMETRIC matrices; MATHEMATICAL mappings; MATRIX rings
- Publication
Acta Mathematica Sinica, 2007, Vol 23, Issue 1, p95
- ISSN
1439-8516
- Publication type
Article
- DOI
10.1007/s10114-005-0770-7