| 二元域上矩阵空间之间的线性秩保持 |
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| 引用本文： | 马立和,张显.二元域上矩阵空间之间的线性秩保持[J].哈尔滨商业大学学报(自然科学版),2005,21(3):336-339. |
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| 作者姓名： | 马立和 张显 |
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| 作者单位： | 黑龙江省计算中心,黑龙江,哈尔滨,150036;黑龙江大学,数学科学学院,黑龙江,哈尔滨,150080 |
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| 基金项目： | This work was supported in part by the Chinese Natural Science Foundation under Grant No.10271021,the Natural Science Foundation of Heilongjiang Province under Grant No.A01-107 and the Fund of Heilongjiang Education Committee for Overseas Scholars under Grant No.1054HQ004. |
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| 摘 要： | 设F2是二元域,n是整数,n≥2.Mn(F2)记F2上的n×n矩阵空间,Sn(F2)记F2上的n×n对称矩阵空间.若线性算子f∶Sn(F2)→Mn(F2)满足rankf(X)=rankX对所有的X∈Sn(F2)成立,则称f是从Sn(F2)到Mn(F2)的线性秩保持.证明了f是从Sn(F2)到Mn(F2)的线性秩保持的充要条件是存在非奇异的U,V∈Mn(F2)满足f∶A→UAV.
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| 关 键 词： | 线性保持 秩 域 对称矩阵 线性空间 |
| 文章编号： | 1672-0946(2005)03-0336-04 |
| 修稿时间： | 2005年1月8日 |
Linear preservers of rank between spaces of matrices over field of two elements |
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| MA Li-he,ZHANG Xian.Linear preservers of rank between spaces of matrices over field of two elements[J].Journal of Harbin University of Commerce :Natural Sciences Edition,2005,21(3):336-339. |
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| Authors: | MA Li-he ZHANG Xian |
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| Institution: | MA Li-he~1,ZHANG Xian~2 |
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| Abstract: | Suppose F2 is the field {0,1} and n is an integer with n≥2.Let Mn(F2) be the linear space of all n×n matrices over F2,and let Sn(F2) be its subspace consisting of all symmetric matrices.A linear operator f:Sn(F2)→Mn(F2) is called a linear preserver of rank from Sn(F2) to Mn(F2) if rank f(X)=rankX for every X∈Sn(F2).It is shown that f is a linear preserver of rank from Sn(F2) to Mn(F2) if and only if there exist nonsingular U,V∈Mn(F2) such that f(A)=UAV for every A∈Sn(F2). |
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| Keywords: | linear preserver rank field symmetric matrix linear space |
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