کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6416886 | 1336897 | 2011 | 8 صفحه PDF | دانلود رایگان |

Let Mn be the algebra of all nÃn matrix over a field F, A a rank one matrix in Mn. In this article it is shown that if a bilinear map Ï from MnÃMn to Mn satisfies the condition that Ï(u,v)=Ï(I,A) whenever u·v=A, then there exists a linear map Ï from Mn to Mn such that Ï(x,y)=Ï(x·y),âx,yâMn. If Ï is further assumed to be symmetric then there exists a matrix B such that Ï(x,y)=tr(xy)B for all x,yâMn. Applying the main result we prove that if a linear map on Mn is desirable at a rank one matrix then it is a derivation, and if an invertible linear map on Mn is automorphisable at a rank one matrix then it is an automorphism. In other words, each rank one matrix in Mn is an all-desirable point and an all-automorphisable point, respectively.
Journal: Linear Algebra and its Applications - Volume 434, Issue 5, 1 March 2011, Pages 1354-1361