A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a...A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a ring R with division ring of quotients, using a new method, and a presentation of GE n(R) is given.展开更多
A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a...A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a ring R with division ring of quotients, using a new method, and a presentation of GE n(R) is given.展开更多
Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and...Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.展开更多
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
We in this paper give a decomposition concerning the general matrix triplet over an arbitrary divisionring F with the same row or column numbers. We also design a practical algorithm for the decomposition of thematrix...We in this paper give a decomposition concerning the general matrix triplet over an arbitrary divisionring F with the same row or column numbers. We also design a practical algorithm for the decomposition of thematrix triplet. As applications, we present necessary and suficient conditions for the existence of the generalsolutions to the system of matrix equations DXA = C1, EXB = C2, F XC = C3 and the matrix equation AXD + BY E + CZF = Gover F. We give the expressions of the general solutions to the system and the matrix equation when thesolvability conditions are satisfied. Moreover, we present numerical examples to illustrate the results of thispaper. We also mention the other applications of the equivalence canonical form, for instance, for the compressionof color images.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
We focus our attention to the set Gr（ξ） of grouplike elements of a coring ξover a ring A. We do some observations on the actions of the groups U（A） and Aut（ξ~） of units of A and of automorphisms of corings of...We focus our attention to the set Gr（ξ） of grouplike elements of a coring ξover a ring A. We do some observations on the actions of the groups U（A） and Aut（ξ~） of units of A and of automorphisms of corings of ξ, respectively, on Gr（ξ）, and on the subset Gal（ξ）of all Galois grouplike elements. Among them, we give conditions on ξ under which Gal（ξ） is a group, in such a way that there is an exact sequence of groups {1} → U（Ag） → U（A） → Gal（ξ） → {1}, where Ag is the subalgebra of coinvaxiants for some g ∈ Gal（ξ）.展开更多
文摘A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a ring R with division ring of quotients, using a new method, and a presentation of GE n(R) is given.
文摘A presentation of hyperbolic unitary group is an important part in the unitary group. The group KG 2,n (R) plays an elementary role in presentation of unitary group. It is proved that KG 2,n(R)=1 for n≥2 over a ring R with division ring of quotients, using a new method, and a presentation of GE n(R) is given.
基金This work is Supported by NSF of Heilongjiang Province
文摘Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.
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 (GrantNo. 60672160)the Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20093108110001)+3 种基金the Scientific Research Innovation Foundation of Shanghai Municipal Education Commission (Grant No. 09YZ13)the Netherlands Organization for Scientific Research (NWO)Singapore MoE Tier 1 Research Grant RG60/07Shanghai Leading Academic Discipline Project (Grant No. J50101)
文摘We in this paper give a decomposition concerning the general matrix triplet over an arbitrary divisionring F with the same row or column numbers. We also design a practical algorithm for the decomposition of thematrix triplet. As applications, we present necessary and suficient conditions for the existence of the generalsolutions to the system of matrix equations DXA = C1, EXB = C2, F XC = C3 and the matrix equation AXD + BY E + CZF = Gover F. We give the expressions of the general solutions to the system and the matrix equation when thesolvability conditions are satisfied. Moreover, we present numerical examples to illustrate the results of thispaper. We also mention the other applications of the equivalence canonical form, for instance, for the compressionof color images.
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Supported by grant MTM2007-61673 from the Ministerio de Educación y Ciencia of SpainP06-FQM-01889 from Junta de Andalucía
文摘We focus our attention to the set Gr（ξ） of grouplike elements of a coring ξover a ring A. We do some observations on the actions of the groups U（A） and Aut（ξ~） of units of A and of automorphisms of corings of ξ, respectively, on Gr（ξ）, and on the subset Gal（ξ）of all Galois grouplike elements. Among them, we give conditions on ξ under which Gal（ξ） is a group, in such a way that there is an exact sequence of groups {1} → U（Ag） → U（A） → Gal（ξ） → {1}, where Ag is the subalgebra of coinvaxiants for some g ∈ Gal（ξ）.
