Norms on complex matrices induced by random vectors II: extension of weakly unitarily invariant norms

TL;DR

This paper simplifies and extends the theory of norms on complex matrices induced by random vectors, particularly classifying weakly unitarily invariant norms and broadening the main theorem's applicability.

Contribution

It provides a simpler proof of the classification of weakly unitarily invariant norms and extends the main theorem to include the case d ≥ 1, removing the need for complex group invariance frameworks.

Findings

Simplified proof of the classification of weakly unitarily invariant norms.

Extended the main theorem to the case d ≥ 1.

Clarified the theoretical foundations of norms on complex matrices.

Abstract

We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to extend the main theorem in [7] from exponent $d\geq 2$ to $d \geq 1$. Our proofs are much simpler than the originals: they do not require Lewis' framework for group invariance in convex matrix analysis. This clarification puts the entire theory on simpler foundations while extending its range of applicability.