# Submultiplicativity of the numerical radius of commuting matrices of order two

## TL;DR

This paper provides an elementary proof that the numerical radius of the product of two commuting 2x2 matrices is at most the product of their numerical radii, including a characterization of cases where equality holds.

## Contribution

It offers a simple proof of submultiplicativity of the numerical radius for commuting 2x2 matrices and characterizes the cases of equality.

## Key findings

- Proof of $w(AB) \,\leq\, w(A)w(B)$ for commuting 2x2 matrices
- Characterization of matrix pairs attaining equality
- Elementary approach to a known inequality

## Abstract

Denote by $w(T)$ the numerical radius of a matrix $T$. An elementary proof is given to the fact that $w(AB) \leq w(A)w(B)$ for a pair of commuting matrices of order two, and characterization is given for the matrix pairs that attain the quality.