Normalitity preserving perturbations and augmentations and their effect on the eigenvalues

TL;DR

This paper investigates how normality-preserving augmentations and perturbations affect the eigenvalues of normal matrices, providing explicit constructions and analyzing eigenvalue perturbations for various matrix classes.

Contribution

It extends previous work by characterizing eigenvalue perturbations due to normality-preserving augmentations and constructs all such augmentations with eigenvalues on quadratic curves.

Findings

Eigenvalues are perturbed predictably by augmentations.

Complete analysis for 2x2 and rank-1 matrices.

Explicit examples illustrating the theoretical results.

Abstract

We revisit the normality preserving augmentation of normal matrices studied by Ikramov and Elsner in 1998 and complement their results by showing how the eigenvalues of the original matrix are perturbed by the augmentation. Moreover, we construct all augmentations that result in normal matrices with eigenvalues on a quadratic curve in the complex plane, using the stratification of normal matrices presented by Huhtanen in 2001. To make this construction feasible, but also for its own sake, we study normality preserving normal perturbations of normal matrices. For $2\times 2$ and for rank-1 matrices, the analysis is complete. For higher rank, all essentially Hermitian normality perturbations are described. In all cases, the effect of the perturbation on the eigenvalues of the original matrix is given. The paper is concluded with a number of explicit examples that illustrate the results and constructions.