Let [Mscr ]n denote the space of complex n×n matrices, and let Ωn denote the spectral unit ball of [Mscr ]n, namely the set of matrices in [Mscr ]n whose eigenvalues lie within the open unit disc. As a step towards the eventual classification of the holomorphic automorphisms of Ωn, we prove that every such automorphism F satisfying F(0) = 0 and F′(0) = I has the property that F(x) is conjugate to x for each x∈Ωn. This result is obtained by combining earlier work of Ransford and White with a general theorem about spectrum-preserving maps proved in this paper.