  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 $S_n(\Omega)$.
