域上迹零矩阵空间上的线性秩1保持

摘要

Suppose F is a field and m is a positive integer with m≥2.Let Mm(F) be the linearspace of all m×m matrices over F,and let slm(F) be the subspace of Mm(F) consisting of all zero-trace matrices.A linear map φ:slm(F)→slm(F) is termed a rank-1 preserver if φ(sl1m(F))(-C)sl1m(F),where sl1m(F) denotes the subset of slm(F) consisting of all rank-1 matrices.It is shown by using an inductive technique that φ:slm(F)→slm(F) is an invertible linear rank-1 preserver if and only if there exist c ∈F* and nonsingular M ∈Mm(F) such thatφ(X)=cMXM-1 for any X ∈slm(F) orφ(X)=cMXT M-1 for any X ∈slm(F).%设F是域,m≥2是正整数,Mn(F)表示域F上所有n×n矩阵构成的线性空间,sln(F)表示Mn(F)的包含所有迹零矩阵的子空间.若线性映射φ:slm(F)→slm(F) 满足φ(sl1m(F))(-C)sl1m(F),则称其为线性秩1保持,其中sl1m(F)定义slm(F)的包含所有秩1矩阵的子集.通过使用数学归纳法证明了:φ:slm(F)→slm(F)是可逆的线性秩l保持的充要条件是存在c ∈F* 和可逆的M ∈Mm(F)使得φ(X)=cMXM-1,(A)X∈slm(F)或φ(X)=cMXT M-1,(A)X ∈slm(F).