矩阵空间上线性保持问题的几个结果

摘要

Let M_n (F) be the linear space of all n×n matrices over a field F,and let sl_n (F) be the subspace of M_n (F) consisting of all zero-trace matrices.Based on some existing results,all invertible linear rank-1 square-zero (respectively,square-zero,involution) preservers on M_n (F) are characterized,and the structures of all strong linear square-zero (respectively,involution) preservers on M_n (F) are described.These results obtained here show clearly the relations among several linear preserver problems.%设M_n(F)表示域F上所有n×n矩阵构成的线性空间,sl_n(F)表示M_n(F)的包含所有迹零矩阵的子空间.基于一些现有的结论,刻划了M_n(F)上可逆的线性秩1平方零(平方零、对合)保持,以及M_n(F)上强线性平方零(对合)保持,所获得的结果展示了几类线性保持问题间的关系.