On the Jordan structure of holomorphic matrices

Abstract

Let $X$ be an open subset of ${\mathbb{C}}^N$, 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. Baumgärtel (Analytic perturbation theory for matrices and operators, Birkhäuser, 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 $X$.