On bilinear maps determined by rank one matrices with some applications

Let Mn be the algebra of all n×n matrix over a field F, A a rank one matrix in Mn. In this article it is shown that if a bilinear map ϕ from Mn×Mn to Mn satisfies the condition that ϕ(u,v)=ϕ(I,A) whenever u·v=A, then there exists a linear map φ from Mn to Mn such that ϕ(x,y)=φ(x·y),∀x,y∈Mn. If ϕ is further assumed to be symmetric then there exists a matrix B such that ϕ(x,y)=tr(xy)B for all x,y∈Mn. 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.