ON THE JORDAN STRUCTURE OF HOLOMORPHIC MATRICES

Let X ⊂ CN be open, and let A be an n × n matrix of holomorphic functions on X. We call a point ξ ∈ X Jordan stable for A if ξ is not a splitting point of the eigenvalues of A and, moreover, there is a neighborhood U of ξ such that, for each 1 ≤ k ≤ n, the number of Jordan blocks of size k in the Jordan normal forms of A(ζ) is the same for all ζ ∈ U. H. Baumg¨artel [4, S 3.4] proved that there is a nowhere dense closed analytic subset of X, which contains the set of all non-Jordan stable points. We give a new proof of this result. This proof shows that the set of non-Jordan stable points ist not only contained in a nowhere dense closed analytic subset, but it is itself such a set, and can be defined by holomorphic functions, the growth of which is bounded by some power (depending only on n) of the growth of A. Also, this proof applies to arbitrary (possibly non-smooth) reduced complex spaces X.