Bilinear transformations on matrices: Rank preservers Original Research Article

Let λ be any element in an algebraically closed field F of characteristic not 2, and let M : Fn × n × Fn × n → Fn × n be a bilinear map on the algebra Fn × n of n × n matrices over F. We prove for n ≥ 5 that if AB + λBA = 0 implies M(A, B) = 0 and rank(AB + λBA) = 1 implies rank M(A, B) = 1 for all matrices A, B set membership, variant Fn × n, then there exist invertible matrices P, Q set membership, variant Fn × n such that either M(X, Y) = P(XY + λYX)Q or M(X, Y) = P(XY + λYX)tQ.