| 矩阵空间之间的秩的线性保持 |
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| 引用本文： | 张显.矩阵空间之间的秩的线性保持[J].黑龙江大学自然科学学报,2005,22(1):12-15. |
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| 作者姓名： | 张显 |
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| 作者单位： | 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： | SupportedbytheNaturalScienceFoundationofChina( 10271021 )，theNaturalScienceFoundationofHeilongjiangProvince(A01 -
07)，FundofHeilongjiangEducationCommitteeforOverseasScholars(1054HQ004) |
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| 摘 要： | 设m,n是正整数,n≥2,F是包含至少三个元素的域.Mn(F)记F上所有n阶矩阵构成的线性空间,Sn(F)记F上所有n阶对称矩阵构成的线性空间.设V和W是Mn(F)的两个子空间.如果线性算子fV→W满足rankf(X)=rankX对于所有的X∈V成立,则称f是从V到W的秩的线性保持.证明了f是从Sn(F)到Mm(F)的秩的线性保持的充分必要条件是n≤m且存在非奇异矩阵U,V∈Mm(F)满足f(A)=U(A+0)V对于所有的A∈Sn(F)成立.由此,确定了所有的从Sn(F)到Sm(F)及从Mn(F)到Mm(F)的秩的线性保持的一般形式.
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| 关 键 词： | 线性保持 秩 矩阵空间 |
Linear preservers of rank between matrix spaces |
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| ZHANG Xian.Linear preservers of rank between matrix spaces[J].Journal of Natural Science of Heilongjiang University,2005,22(1):12-15. |
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| Authors: | ZHANG Xian |
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| Abstract: | Suppose m,n are positive integers with n≥2, and F is any field except for {0,1}. Let Mn(F) be the linear space of all n × n matrices over F,and let Sn(F) be its subspace consisting of all symmetric matrices. For two subspaces V and W of Mn(F),a linear operator f: V→W is called a linear preserver of rank from V to W if rankf(X) = rankX for every X ∈ V. It is shown that f is a linear preserver of rank from Sn (F) to Mm (F) if and only if n ≤ m and there exist nonsingular matrices U, V ∈Mm (F)such thatf(A)= U(A0) V for every A ∈Sn(F). Thereby, the general form of all linear preservers of rank from Sn(F)(respectively, Mn(F)) to Sm(F) (respectively, Mm(F)) is determined. |
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| Keywords: | linear preserver rank matrix space |
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