A local form for the automorphisms of the spectral unit ball

If $F$ is an automorphism of $\Omega_n$, the $n^2$-dimensional spectral unit ball, we show that, in a neighborhood of any cyclic matrix of $\Omega_n$, the map $F$ can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result.