矩阵代数上的局部合同变换和局部相似变换

令F表示任意域,Mn（F）表示由F上所有n×n矩阵形成的结合代数.本文的目的是研究Mn（F）上具有如下性质的两类线性映射,其中一类线性映射在Mn（F）上每一点的取值与Mn（F）的某个合同变换在该点的取值相同,另一类线性映射在Mn（F）上每一点的取值与Mn（F）的某个相似变换在该点的取值相同,随着Mn（F）上的点不同,这些合同变换和相似变换可能也不同.利用矩阵的秩、幂等阵以及幂零阵的性质,通过矩阵计算的方法证明了第一类线性映射或者是合同变换或者是合同变换与转置变换的复合,第二类线性映射或者是相似变换或者是相似变换与转置变换的复合.由这个结果可知存在真正意义上的局部合同变换和局部相似变换,从而丰富了局部映射理论的研究。

Local congruence transformations and local similarity transformations of matrix algebras

Let F be an arbitrary filed,Mn（F） the algebra of all n by n matrices over F.The aim of this paper is to describe the linear map of Mn（F） which at each point is equal to the value of some congruence transformation of Mn（F） and the linear map which agrees at each point with some similarity transformation,where the congruence transformation and the similarity transformation may be different from point to point.Using the property of matrix’s rank and idempotent,by the skill of matrix computation,it is proved that the first kind map is either a congruence transformation or a congruence transformation composed with a transpo-sition,the second kind map is either a similarity transformation or a similarity transformation composed with a transposition,from this result it is known that there do exists proper local congruence transformation and local similarity transformation.This result enriches the theory of local maps.