Normalitity preserving perturbations and augmentations and their effect on the eigenvalues
This is an incremental theoretical contribution for mathematicians studying matrix perturbations and eigenvalue behavior.
The paper revisits normality-preserving augmentations of normal matrices, showing how eigenvalues are perturbed, and constructs all augmentations yielding normal matrices with eigenvalues on a quadratic curve. It provides complete analysis for 2x2 and rank-1 matrices, and describes essentially Hermitian normality perturbations for higher rank.
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.