| 域上偶数阶交错矩阵空间的加法秩保持 |
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| 引用本文： | 张显,袁晖坪.域上偶数阶交错矩阵空间的加法秩保持[J].黑龙江大学自然科学学报,2005,22(5). |
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| 作者姓名： | 张显 袁晖坪 |
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| 作者单位： | 1. 黑龙江大学,数学科学学院,黑龙江,哈尔滨,150080
2. 重庆工商大学,理学院,重庆,400067 |
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| 基金项目： | SupportedbytheNaturalScienceFoundationofChina(10271021)，theNaturalScienceFoundationofHeilongjiangProvince(A01-07)，theFundofHeilongjiangEducationCommitteeforOverseasScholars(1054HQ004) |
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| 摘 要： | 设Kn(F)是域F上所有n×n交错矩阵构成的线性空间.如果一个算子f:Kn(F)→Kn(F)满足对所有的A,B∈Kn(F)有f(A+B)=f(A)+f(B)并且对任意的X∈Kn(F)有rankf(X)=rankX,则称f是Kn(F)上的加法秩保持.当n是不小于4的整数且F任意时,证明了f是Kn(F)上的加法秩保持当且仅当存在非零的纯量γ、非奇异的n×n矩阵P和域F的单自同态δ满足或者f:aij]|→αPaijδ]PT,或者n=4且f:aij]|→αP★(aiδj])PT,其中★:K4(F)→K4(F)表示对换(1,4)和(2,3)位置元素及(4,1)和(3,2)位置元素的算子.
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| 关 键 词： | 域 秩 加法保持 交错矩阵 |
Additive rank preservers on n × n alternate matrix spaces over fields:the case that n is even |
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| ZHANG Xian,YUAN Hui-ping.Additive rank preservers on n × n alternate matrix spaces over fields:the case that n is even[J].Journal of Natural Science of Heilongjiang University,2005,22(5). |
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| Authors: | ZHANG Xian YUAN Hui-ping |
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| Institution: | ZHANG Xian~1,YUAN Hui-ping~2 |
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| Abstract: | Let Kn (F) be the linear space of all n × n alternate matrices over a field F. An operator f:Kn(F)→Kn(F) is called an additive preserver of rank on Kn(F) iff(A +B) =f(A) +f(B) for all A,B ∈ Kn (F) and rank f(X) = rankX for every X ∈ Kn(F). It is shown that, assuming that n is even with n ≥4 and F is arbitrary, f is an additive preserver of rank on Kn (F) if, and only if, there exist a nonzero scalar γ, a nonsingular n × n matrix P and an injective field endomorphism δ of F such that either f: aij]|→αP aδij]PT, or n = 4 and f: aij ] |→αP ★ ( aδij] ) pT, where ★: K4 ( F ) →K4 (F) denotes the operator exchanging the (1,4) -th entry with the (2,3) -th entry and the (4,1) -th entry with the (3,2) -th entry. |
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| Keywords: | field rank additive preserver alternate matrix |
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