On the Jordan structure of holomorphic matrices

TL;DR

This paper provides a new, more precise proof that the set of points where a holomorphic matrix is not Jordan stable forms a nowhere dense analytic subset, extending previous results to more general complex spaces.

Contribution

It offers a novel proof of the structure of non-Jordan stable points for holomorphic matrices, with improved precision and applicability to non-smooth complex spaces.

Findings

The set of non-Jordan stable points is a nowhere dense analytic subset.

The proof applies to arbitrary complex spaces, including non-smooth ones.

Provides estimates and a more precise description of the stability set.

Abstract

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 $X$.