On minimal norms on $M_n$

TL;DR

This paper characterizes minimal matrix norms on complex matrices as maximums over vector norms, extending known results on algebra norms.

Contribution

It demonstrates that every minimal norm on $M_n$ can be represented via vector norms on ${f C}^n$, generalizing previous characterizations.

Findings

Every minimal norm on $M_n$ can be expressed as a maximum over vector norms.

The result extends known characterizations of algebra norms.

Provides a new perspective on the structure of minimal matrix norms.

Abstract

In this note, we show that for each minimal norm $N(\cdot)$ on the algebra $M_n$ of all $n \times n$ complex matrices, there exist norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on ${\mathbb C}^n$ such that $$N(A)=\max\{\|Ax\|_2: \|x\|_1=1, x\in {\mathbb C}^n\}$$ for all $A \in M_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.