The Gau-Wu Number for $4\times 4$ and Select Arrowhead Matrices

TL;DR

This paper introduces dichotomous matrices as a generalization of essentially Hermitian matrices, establishes criteria for arrowhead matrices to be dichotomous and unitarily irreducible, and computes the Gau-Wu number for certain classes, fully classifying 4x4 matrices by this measure.

Contribution

It defines dichotomous matrices, provides criteria for arrowhead matrices to be dichotomous and unitarily irreducible, and classifies all 4x4 matrices based on their Gau-Wu numbers.

Findings

Criteria for arrowhead matrices to be dichotomous established

Complete classification of 4x4 matrices by Gau-Wu number achieved

Gau-Wu number computed for a broad class of arrowhead matrices

Abstract

The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau--Wu number (i.e., the maximal number $k(A)$ of orthonormal vectors $x_j$ such that the scalar products $\langle Ax_j,x_j\rangle$ lie on the boundary of the numerical range of $A$) is computed for a class of arrowhead matrices $A$ of arbitrary size, including dichotomous ones. These results are then used to completely classify all $4\times4$ matrices according to the values of their Gau--Wu numbers.