Let A be a holomorphic \(n\times n\)-matrix function on a region \(G\subseteq {\mathbb{C}}\). Under what conditions does there exist a holomorphic matrix function S on G with invertible values such that \(S^{-1}(z)A(z)S(z)\) is a Jordan-matrix for all \(z\in G?\)
A necessary condition is, obviously, that the Jordan-form of A(z) can be written as a holomorphic matrix function, but this is not sufficient, unless the number of different eigenvalues of A(z) is constant on G. This latter condition can be generalized, and the former be weakened, in order to obtain necessary and sufficient conditions. These conditions are of a local nature, and the set where they are not met is discrete in G. Improving a result of H. Baumgärtel it is shown that A is always meromorphically similar to a holomorphic Jordan form \(J=S^{-1}AS\) on a Riemann surface M over \(G\setminus \gamma\), \(\gamma\) discrete, and that the set of poles of S, \(S^{-1}\) is characterized as the set of points in M where the abovementioned conditions fail.