Matricial Radius: A Relation of Numerical radius with Matricial Range

TL;DR

This paper introduces a new relation called Matricial Radius that connects the numerical radius of a matrix with its matricial range, providing new insights into matrix analysis.

Contribution

It establishes a novel relation between the numerical radius and the matricial range of matrices, expanding understanding of matrix spectral properties.

Findings

Derived a formula linking numerical radius and matricial range

Showed equivalence of different supremum characterizations

Provided theoretical insights into matrix spectral analysis

Abstract

It has been shown that if $T$ is a complex matrix, then {\small\begin{align*} \omega(T)&=\frac{1}{n}\sup\left\{|\mathrm{Tr}\ X|;\ X\in W^n(T)\right\}\\ &=\frac{1}{n}\sup\left\{\|X\|_1;\ X\in W^n(T)\right\}\\ &= \sup\left\{ \omega(X);\ X\in W^n(T)\right\} \end{align*} } where $n$ is a positive integer, $\omega(T)$ is the numerical radius and $W^n(T)$ is the $n$'th matricial range of $T$.