Maps on matrices that preserve the spectral radius distance

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