A Resolvent Criterion for Normality

TL;DR

This paper establishes new criteria for normality of matrices using pseudospectra and resolvent norms, showing that spectral properties can determine normality and providing practical tests for it.

Contribution

It introduces a resolvent-based criterion that is both necessary and sufficient for normality, linking spectrum and spectral norm in a novel way.

Findings

A distance formula for normality is proven to be both necessary and sufficient.

Spectral norm of a matrix's resolvent can be exactly recovered from its spectrum if the matrix is normal.

New criteria for normality are derived from pseudospectra and resolvent behavior.

Abstract

Given a normal matrix $A$ and an arbitrary square matrix $B$ (not necessarily of the same size), what relationships between $A$ and $B$, if any, guarantee that $B$ is also a normal matrix? We provide an answer to this question in terms of pseudospectra and norm behavior. In doing so, we prove that a certain distance formula, known to be a necessary condition for normality, is in fact sufficient and demonstrates that the spectrum of a matrix can be used to recover the spectral norm of its resolvent precisely when the matrix is normal. These results lead to new normality criteria and other interesting consequences.