On bilinear maps determined by rank one matrices with some applications (Q624539)

This paper was inspired by recently appeared several papers which describe bilinear maps on algebras through their action on elements whose products are zero. Let \(M_n\) be the algebra of all \(n\times n\) matrices over a field \(F\), \(A\) a rank one matrix in \(M_n\). The authors show that if a bilinear map \(\phi\), from \(M_n\times M_n\) to \(M_n\) satisfies the condition that \(\phi(u,v)= \phi(I,A)\) whenever \(u\cdot v= A\), then there exists a linear map \(\varphi\) from \(M_n\) to \(M_n\) such that \(\phi(x,y)= \varphi(x\cdot y)\), \(\forall x\), \(y\in M\). If \(\phi\) is further assumed to be symmetric then there exists a matrix \(B\) such that \(\phi(x,y)= \text{tr}(xy)B\) for all \(x,y\in M_n\). Applying the main result the paper proves 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 automorphizable 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-automorphizable point, respectively.