How well-conditioned can the eigenvalue problem be?

TL;DR

This paper investigates the minimal possible condition number for eigenvalue problems, providing an exact first-order asymptotic analysis to understand how well-conditioned such problems can be.

Contribution

It offers a precise asymptotic characterization of the smallest achievable condition number for eigenvalue computations, addressing a fundamental question in numerical linear algebra.

Findings

Identifies the lower bound of the eigenvalue condition number asymptotically.

Provides an exact first-order asymptotic formula for the minimal condition number.

Enhances understanding of the intrinsic conditioning limits of eigenvalue problems.

Abstract

The condition number for eigenvalue computations is a well--studied quantity. But how small can we expect it to be? Namely, which is a perfectly conditioned matrix w.r.t. eigenvalue computations? In this note we answer this question with exact first order asymptotic.