  Let $X$ be an open subset of $\Bbb C^N$, and let $A$ be an $n\times n$ matrix
of holomorphic functions on $X$. We call a point $\xi\in X$ $\mathbf{Jordan}$
$\mathbf{stable}$ for $A$ if $\xi$ is not a splitting point of the eigenvalues
of $A$ and, moreover, there is a neighborhood $U$ of $\xi$ such that, for each
$1\le k\le n$, the number of Jordan blocks of size $k$ in the Jordan normal
forms of $A(\zeta)$ is the same for all $\zeta\in U$. H. Baumg\"artel (Analytic
perturbation theory for matrices and operators, Birkh\"auser, 1985) proved that
there is a nowhere dense closed analytic subset of $X$, which contains all
points of $X$ which are not Jordan stable for $A$. We give a new proof of this
result. This proof has the advantage that the result can be obtained in a more
precise form, and with some estimates. Also, this proof applies to arbitrary,
possibly non-smooth, complex spaces $X$.

Complex Variables