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 ...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.展开更多
Let D be a division ring with an involution ?, $ \mathcal{H}_2 $ (D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A ? B) be the arithmetic distance between A, B ∈ $ \mathcal{H}_2 $ (D). In t...Let D be a division ring with an involution ?, $ \mathcal{H}_2 $ (D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A ? B) be the arithmetic distance between A, B ∈ $ \mathcal{H}_2 $ (D). In this paper, the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D (char(D) ≠ = 2) is proved: if φ: $ \mathcal{H}_2 $ (D) → $ \mathcal{H}_2 $ (D) is the adjacency preserving bijective map, then φ is of the form φ(X) = $ ^t \bar P $ X σ P +φ(0), where P ∈ GL 2(D), σ is a quasi-automorphism of D. The quasi-automorphism of D is studied, and further results are obtained.展开更多
Let R be a unital*-ring.For any a,w,b∈R,we apply the w-core inverse to define a new class of partial orders in R,called the w-core partial order.Suppose that a,b∈R are w-core invertible.We say that a is below b unde...Let R be a unital*-ring.For any a,w,b∈R,we apply the w-core inverse to define a new class of partial orders in R,called the w-core partial order.Suppose that a,b∈R are w-core invertible.We say that a is below b under the w-core partial order if a_(w)^(#)a=a_(w)^(#)b and a_(w)a_(w)^(#)=bwa_(w)^(#),where a_(w)^(#)denotes the w-core inverse of a.Characterizations of the w-core partial order are given,and its relationships with several types of partial orders are also considered.In particular,we show that the core partial order coincides with the a-core partial order,and the star partial order coincides with the a*-core partial order.展开更多
基金the National Natural Science Foundation of China Grant,#10271021
文摘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.
基金supported by National Natural Science Foundation of China (Grant No. 10671026)
文摘Let D be a division ring with an involution ?, $ \mathcal{H}_2 $ (D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A ? B) be the arithmetic distance between A, B ∈ $ \mathcal{H}_2 $ (D). In this paper, the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D (char(D) ≠ = 2) is proved: if φ: $ \mathcal{H}_2 $ (D) → $ \mathcal{H}_2 $ (D) is the adjacency preserving bijective map, then φ is of the form φ(X) = $ ^t \bar P $ X σ P +φ(0), where P ∈ GL 2(D), σ is a quasi-automorphism of D. The quasi-automorphism of D is studied, and further results are obtained.
基金The authors are highly grateful to the referees for their valuable comments and suggestions which greatly improved this paper.This research is supported by the National Natural Science Foundation of China(No.11801124)China Postdoctoral Science Foundation(No.2020M671068).
文摘Let R be a unital*-ring.For any a,w,b∈R,we apply the w-core inverse to define a new class of partial orders in R,called the w-core partial order.Suppose that a,b∈R are w-core invertible.We say that a is below b under the w-core partial order if a_(w)^(#)a=a_(w)^(#)b and a_(w)a_(w)^(#)=bwa_(w)^(#),where a_(w)^(#)denotes the w-core inverse of a.Characterizations of the w-core partial order are given,and its relationships with several types of partial orders are also considered.In particular,we show that the core partial order coincides with the a-core partial order,and the star partial order coincides with the a*-core partial order.
