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.