On low rank perturbation of matrices

TL;DR

This paper explores how low-rank perturbations can significantly alter matrix spectra and forms, highlighting differences between general and normal matrices, and analyzing the proximity of matrices to unitarity.

Contribution

It provides new insights into the effects of low-rank perturbations on spectra and Jordan forms, especially for normal matrices and the near-unitary property.

Findings

Rank 1 perturbations can drastically change simple spectra.

Normal matrices exhibit different perturbation behavior compared to general matrices.

Almost unitary matrices are close to truly unitary matrices in rank distance.

Abstract

The article is devoted to different aspects of the question "What can be done with a matrix by low rank perturbation?" It is proved that one can change a geometrically simple spectrum drastically by a rank 1 permutation, but the situation is quite different if one restricts oneself to normal matrices. Also, the Jordan normal form of a perturbed matrix is considered. It is proved that with respect to rank as a distance all almost unitary matrices are near unitary.