Eigenvalue continuity and Ger\v{s}gorin's theorem

TL;DR

This paper clarifies the different notions of eigenvalue continuity and revisits Geršgorin's disk theorem to address ambiguities in its proof related to continuity assumptions.

Contribution

It provides a clear distinction between types of eigenvalue continuity and clarifies the proof of Geršgorin's theorem regarding these continuity concepts.

Findings

Clarification of eigenvalue continuity types

Analysis of Geršgorin's theorem proof

Resolution of proof ambiguities

Abstract

Two types of eigenvalue continuity are commonly used in the literature. However, their meanings and the conditions under which continuities are used are not always stated clearly. This can lead to some confusion and needs to be addressed. In this note, we revisit the Ger\v{s}gorin disk theorem and clarify the issue concerning the proofs of the theorem by continuity.