On a spectral version of Cartan's theorem

For a domain $\Omega$ in the complex plane, we consider the domain $S_n(\Omega)$ consisting of those $n\times n$ complex matrices whose spectrum is contained in $\Omega$. Given a holomorphic self-map $\Psi$ of $S_n(\Omega)$ such that $\Psi(A)=A$ and the derivative of $\Psi$ at $A$ is identity for some $A\in S_n(\Omega)$, we investigate when the map $\Psi$ would be spectrum-preserving. We prove that if the matrix $A$ is either diagonalizable or non-derogatory then for most domains $\Omega$, $\Psi$ is spectrum-preserving on $S_n(\Omega)$. Further, when $A$ is arbitrary, we prove that $\Psi$ is spectrum-preserving on a certain analytic subset of $S_n(\Omega)$.