保持矩阵迹的反乘法映射

Anti-Multiplicative Maps on Matrices that Preserve the Trace

设P是一个域,Г是满足{aEij︱i,j=2,…,n,a∈P}ГMn(P)的一个乘法半群,其中Mn(P)定义P上所有n×n矩阵组成的乘法半群。本文证明了一个结果:若f:Г→Mn(P)是一个保迹反乘法映射,则存在可逆矩阵S∈Mn(P),使得f(A)=SATS-1,A∈Г。由此刻画了Г的保迹反乘法映射。

Suppose P is a field, let F be multiplicative semigroup which satisfies {aEy｜i,j=2,…,n,a∈P} lohtain in Г lohtain in Mn（P）, where Mn（P） denotes the semigroup of all n x n matrices over P. In this paper, we prove a result： suppose f：Г→Mn（P）is a anti -multiplicative map that preserve trace, then there exists an inverti- ble S∈Mn（P） which form f（A）=SA^TS^-1,YA∈Г.