On bilinear maps determined by rank one matrices with some applications

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• 作者：Long Wang^a ; ^{wanglongxuzhou@126.com} ; Yizheng Fan^a ; Xiaobin Ma^b
• 关键词：Bilinear maps ; All-desirable points ; All-automorphisable points
• 刊名：Linear Algebra and its Applications
• 出版年：2011
• 出版时间：1 March 2011
• 年：2011
• 卷：434
• 期：5
• 页码：1354-1361
• 全文大小：183 K

文摘

Let M_n be the algebra of all n×n matrix over a field F, A a rank one matrix in M_n. In this article it is shown that if a bilinear map from M_n×M_n to M_n satisfies the condition that (u,v)=(I,A) whenever u·v=A, then there exists a linear map φ from M_n to M_n such that . If is further assumed to be symmetric then there exists a matrix B such that (x,y)=tr(xy)B for all x,yM_n. Applying the main result we prove that if a linear map on M_n is desirable at a rank one matrix then it is a derivation, and if an invertible linear map on M_n is automorphisable at a rank one matrix then it is an automorphism. In other words, each rank one matrix in M_n is an all-desirable point and an all-automorphisable point, respectively.