Max-min and min-max approximation problems for normal matrices revisited

TL;DR

This paper presents a new proof for an equality in max-min and min-max approximation problems involving normal matrices, connecting matrix approximation problems with classical complex approximation theory.

Contribution

It introduces a novel proof leveraging classical approximation theory, simplifying the understanding of the relationship between matrix and complex set approximation problems.

Findings

Established a new proof for the approximation equality

Linked matrix approximation problems to classical complex approximation

Simplified the theoretical framework using classical theorems

Abstract

We give a new proof for an equality of certain max-min and min-max approximation problems involving normal matrices. The previously published proofs of this equality apply tools from matrix theory, (analytic) optimization theory and constrained convex optimization. Our proof uses a classical characterization theorem from approximation theory and thus exploits the link between the two approximation problems with normal matrices on the one hand and approximation problems on compact sets in the complex plane on the other.