Normal matrices

TL;DR

This paper characterizes normal matrices by examining the spectral norm distances from a matrix to matrices with specific eigenvalues, establishing a criterion based on equality of these distances for multiple points.

Contribution

It introduces a new characterization of normal matrices using spectral norm distances to matrices with given eigenvalues based on spectral properties.

Findings

Equality of spectral norm distances for many points implies normality.

Provides a spectral norm-based criterion for normal matrices.

Connects eigenvalue proximity with matrix normality.

Abstract

Let $A$ be a square complex matrix and $z$ a complex number. The distance, with respect to the spectral norm, from $A$ to the set of matrices which have $z$ as an eigenvalue is less than or equal to the distance from $z$ to the spectrum of $A$. If these two distances are equal for a sufficiently large finite set of numbers $z$ which are not in the spectrum of $A$, then the matrix $A$ is normal.