保持秩1矩阵的加法映射  

Additive Maps Preserving Rank-one Matrices

作　　者：寻杨[1] 马立和[2] 张显[3] 

机构地区：[1]济宁师专数学系,山东济宁272125 [2]黑龙江省计算中心,黑龙江哈尔滨150000 [3]黑龙江大学数学科学学院,黑龙江哈尔滨150080

出　　处：《莆田学院学报》2005年第2期1-5,共5页Journal of putian University

基　　金：国家自然科学基金资助项目(10271021);黑龙江省自然科学基金资助项目(A01-07);黑龙江省教育厅海外学人资助项目(1054HQ004)

摘　　要：设m、n、p、q是正整数,F是不同构于它自身的真子域的域,Mmn(F)记F上所有m×n矩阵的集合,M1mn(F)记Mm(nF)的包含所有秩1矩阵的子集。若一个映射f:Mm(nF)→Mpq(F)满足f(M1mn(F))哿M1pq(F)且f(A+B)=f(A)+f(B),坌A,B∈Mmn(F),则称f是保持秩1矩阵的加法映射。证明了:若一个保持秩1矩阵的加法映射f:Mm(nF)→Mp(qF)满足存在G,H∈Mm1n(F)使得rank(f(G)+f(H))>1,则存在P∈GL(pF),Q∈GL(qF)和F的域自同构啄使得1)p叟m叟2,q叟n叟2,f:A｜→P(A啄堠0)Q;或者2)p叟n叟2,q叟m叟2,f:A｜→P((A啄)T堠0)Q。Suppose m, n, p, q are positive integers and F is a field that is not isomorphic to a proper subfield of itself. Let Mmn(F) be the set of all m×n matrices over F and let M (F) be its subset consisting of all rank one matrices. A map f: Mmn(F)→Mpq(F) is said to be an additive rank-one preserver if f (M (F))→M ( F) and f (A+B) = f (A) + f (B) for any A,B∈Mmn(F). In this paper it is shown that if f :Mmn(F)→Mpq(F) is an additive map preserving rank-one matrices and satisfying rank(f (G) + f (H)) >1 for some G,Hspillway; reservoir; project design M (F), then there are P∈GLp(F),Q∈GLq(F) and a field automorphism δ on F such that either 1) p≥m≥2, q≥n≥2 and f :A|→P(Aδ 0)Q; or 2) p≥n≥2, q≥m≥2 and f:A|→P((Aδ)T 0)Q.

关 键 词：加法映射 正整数 N矩阵 自同构 集合 子集 AA 

分 类 号：O153.3[理学—数学] O156.1[理学—基础数学]