二元域上矩阵空间之间的线性秩保持  

Linear preservers of rank between spaces of matricesover field of two elements

作　　者：马立和[1] 张显[2] 

机构地区：[1]黑龙江省计算中心,黑龙江哈尔滨150036 [2]黑龙江大学数学科学学院,黑龙江哈尔滨150080

出　　处：《哈尔滨商业大学学报（自然科学版）》2005年第3期336-339,共4页Journal of Harbin University of Commerce：Natural Sciences Edition

基　　金：supported in part by the Chinese Natural Sciencie Foundation under Grant No.10271021;the Natural Science Foundation of Heilongjiang Province under Grant No.A01-107;the Fund of Heilongjiang Education Committee for Overseas Scholars under Grant No.1054HQ004.

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.Suppose F_2 is the field {0,1} and n is an integer with n≥2.Let M_n(F_2) be the linear space of all n×n matrices over F_2,and let S_n(F_2) be its subspace consisting of all symmetric matrices.A linear operator f:S_n(F_2)→M_n(F_2) is called a linear preserver of rank from S_n(F_2) to M_n(F_2) if rank f(X)=rankX for every X∈S_n(F_2).It is shown that f is a linear preserver of rank from S_n(F_2) to M_n(F_2) if and only if there exist nonsingular U,V∈M_n(F_2) such that f(A)=UAV for every A∈S_n(F_2).

关 键 词：线性保持 秩 域 对称矩阵 线性空间 

分 类 号：O151.21[理学—数学]