Explicit Bounds for the Pseudospectra of Various Classes of Matrices and Operators

TL;DR

This paper provides explicit bounds and characterizations for the pseudospectra of various matrix classes, including 2x2 matrices, bidiagonal matrices, and finite rank operators, enhancing understanding of their spectral stability.

Contribution

It offers a complete characterization of 2x2 matrix pseudospectra and asymptotic analysis for general matrices, along with explicit bounds for bidiagonal and finite rank operators.

Findings

Complete characterization of 2x2 matrix pseudospectra

Asymptotic behavior of pseudospectra as epsilon approaches zero

Explicit bounds for bidiagonal matrices and finite rank operators

Abstract

We study the $\epsilon$-pseudospectra $\sigma_\epsilon(A)$ of square matrices $A \in \mathbb{C}^{N \times N}$. We give a complete characterization of the $\epsilon$-pseudospectrum of any $2 \times 2$ matrix and describe the asymptotic behavior (as $\epsilon \to 0$) of $\sigma_\epsilon(A)$ for any square matrix $A$. We also present explicit upper and lower bounds for the $\epsilon$-pseudospectra of bidiagonal matrices, as well as for finite rank operators.