Nonlinear determinant preserving maps on matrix algebras

摘要

Let F be a field having at least n(2) + 1 distinct elements, and denote by M-n the space of all n x n matrices over F. We prove that if phi and psi are maps from M-n into itself, one of them being surjective, such that det(phi(x) + psi(y)) = det(x + y) for all x, y is an element of M-n, there exist then x(0) is an element of M-n, and u, v is an element of M-n satisfying det(uv) = 1 such that either phi(x) = u(x + x(0))v and psi(x) = u(x - x(0))v for all x is an element of M-n, or phi(x) = u(x + x(0))(t)v and psi(x) = u(x - x(0))(t)v for all x is an element of M-n. (C) 2019 Elsevier Inc. All rights reserved.