Rank one perturbations of H-positive real matrices

We consider a generic rank one structured perturbation on H-positive real matrices. The case with complex rank one perturbation is treated in general, but the main focus of this article is the real rank one perturbation. In general, the H-positive real matrix A which is given in Jordan canonical form loses the largest Jordan block after a rank one perturbation for each eigenvalue. Surprisingly, for a real H-skew symmetric matrix for which the largest Jordan block at eigenvalue zero has even size and for a real H-nonnegative rank one perturbation the largest Jordan block with zero eigenvalue grows one in size. Generic Jordan structures of perturbed matrices are identified.