ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

## 20th ILAS Conference

In minisymposium: Plenary Talks

Tue 08:30–09:15, Auditorium Max Weber, Chair: Hugo Woerdeman
Eigenvalue perturbation theory of classes of structured matrices under generic structured small rank perturbations
André Ran (VU Amsterdam)

In this talk the perturbation theory of structured matrices under structured small rank perturbations will be discussed. Several classes of structured matrices will be considered, among them, $J$-Hamiltonian ($JA=-A^TJ$ for a skew-symmetric $J$), $H$-orthogonal ($A^THA=H$ for a symmetric $H$), but also $H$-positive real ($HA+A^TH \geq0$) and $H$-expansive matrices ($A^THA -H \geq 0$).

We shall discuss the behaviour of the eigenvalues under generic structured small rank perturbations; for instance, when $A$ is $H$-orthogonal, we shall consider the eigenvalues of $B(t)=(I-\frac{2t}{u^THu}uu^TH)A$ where the vector $u$ is generic.

Generic Jordan structures of perturbed matrices are identified. It is shown in joint work with L. Batzke, Chr. Mehl, V. Mehrmann and L. Rodman [4,6,1] that in many cases the perturbation behavior of the Jordan structures in the case of $J$-Hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in [3,7,8,9].

We shall discuss also $H$-skew-symmetric matrices. A rank one perturbation of the form $B=A+uu^*H$ is not in the class of $H$-skew-symmetric matrices, however, it is a so-called $H$-positive real matrix. A-typical behaviour similar to the $J$-Hamiltonian case occurs also in this case. In addition, the eigenvalues of $B$ which are not also eigenvalues of $A$ are necessarily off the imaginary axis. These results arise as a special case of the study of low rank perturbations of $H$-positive real matrices. This is joint work with D.B. Janse van Rensburg, J.H. Fourie and G.J. Groenewald, [2]. Recent results for $H$-expansive perturbations of $H$-orthogonal matrices will be discussed as well.

1. L. Batzke, Chr. Mehl, A.C.M. Ran and L. Rodman: Generic rank-$k$ perturbations of structured matrices. To appear in Proceedings IWOTA 2014, Birkhäuser (2016).
2. J.H. Fourie, G.J. Groenewald, D.B. Janse van Rensburg, and A.C.M. Ran: Rank one perturbations of $H$-positive real matrices. Linear Algebra and its Applications 439 (2013), 653–674.
3. L. Hörmander and A. Melin, A remark on perturbations of compact operators, Math. Scand. 75 (1994) 255–262.
4. C. Mehl, V. Mehrmann, A.C.M. Ran and L. Rodman: Eigenvalue perturbation theory of structured matrices under generic structured rank one perturbations. Linear Algebra and its Applications, 435 (2011), 687–716.
5. C. Mehl, V. Mehrmann, A.C.M. Ran, L. Rodman: Perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations. Linear Algebra and its Applications 436 (2012), 4027-4042.
6. C. Mehl, V. Mehrmann, A.C.M. Ran, L. Rodman: Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations. BIT Numerical Mathematics 54 (2014), 219–255.
7. J. Moro and F. Dopico, Low rank perturbation of Jordan structure, SIAM J. Matrix Analysis and Appl. 25 (2003) 495–506.
8. S.V. Savchenko, Typical changes in spectral properties under perturbations by a rank-one operator, Mat. Zametki 74 (2003) 590–602.
9. S.V. Savchenko, On the change in the spectral properties of a matrix under a perturbation of a sufficiently low rank, Funktsional. Anal. i Prilozhen 38 (2004) 85–88.