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

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).