Maps on matrices that preserve the spectral radius distance

Let phi be a surjective map on the space of n x n complex matrices such tha t r(phi(A) - phi(B)) = r(A - B) for all A, B, where r(X) is the spectral ra dius of X. We show that phi must be a composition of five types of maps: tr anslation, multiplication by a scalar of modulus one, complex conjugation, taking transpose and (simultaneous) similarity. In particular, phi is real linear up to a translation.