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

TL;DR

This paper studies the behavior of eigenvalues of rank-one perturbations of matrices, exploring their global properties, limits, and analytic formulas across various matrix classes including unstructured and structured types.

Contribution

It provides new insights into the eigenvalue behavior of rank-one perturbations for multiple matrix classes, including conditions for global analytic formulas and eigenvalue limits.

Findings

Eigenvalues exhibit specific global behaviors under rank-one perturbations.

Limits of eigenvalues as perturbation parameter tends to infinity are characterized.

Analytic formulas for eigenvalues are established for certain matrix classes.

Abstract

General properties of eigenvalues of $A+\tau uv^*$ as functions of $\tau\in\Comp$ or $\tau\in\Real$ or $\tau=\e^{\ii\theta}$ 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.