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

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.,...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2021-05
Main Authors: Camenga, Kristin A, Rault, Patrick X, Spitkovsky, Ilya M, Johnson Yates, Rebekah B
Format: Article
Language:English
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422