Global properties of eigenvalues of parametric rank one perturbations for unstructured and structured matrices

Abstract

General properties of eigenvalues of $A+\tau u{v}^{\ast }$ as functions of $τ\in\Comp$ or $τ\in\Real$ or $τ=\e^{\iiθ}$ on the unit circle are considered. In particular, the problem of existence of global analytic formulas for eigenvalues is addressed. Furthermore, the limits of eigenvalues with $\tau \to \infty$ are discussed in detail. The following classes of matrices are considered: complex (without additional structure), real (without additional structure), complex $H$-selfadjoint and real $J$-Hamiltonian.