Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations

Abstract

An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a crucial feature of matrices in these classes. The behaviour of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Applications include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients.