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

General properties of eigenvalues of $A+τuv^*$ 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 $τ\to\infty$ 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.

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