## General Mathematics Colloquium

**18May2016**11:15 - 12:15

Speaker: André Ran (VU and NWU South Africa)

Title: Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations

### Abstract

Perturbation theory for eigenvalues and Jordan normal forms of matrices will be discussed, in particular the following question: given a square matrix A, what can be said about the Jordan normal form of A+B, where B has rank one? The interest is not so much in what can happen, but in what happens "mostly", i.e., for generic rank one perturbations B.

The unstructured case will be discussed first. The main result goes back to Hörmander and Melin (1994), and was rediscovered about a decade later by Dopico and Moro and by Savchenko.

Matrices which have structure with respect to an indefinite inner product appear in many problems in applications: Hamiltonian matrices and symplectic matrices turn up in e.g., applications in systems and control engineering. For such matrices rank one perturbations which are also structured are discussed. Compared to the unstructured case there are some surprising differences.

The results which will be presented are part of joint work with Chr. Mehl, V. Mehrmann and L. Rodman.

Location: KdVI meeting room, SP 105-107, F3.20