On the Phases of a Semi-Sectorial Matrix
Abstract
In this paper, we extend the definition of phases of sectorial matrices to those of semi-sectorial matrices, which are possibly singular. Properties of the phases are also extended, including those of the Moore-Penrose generalized inverse, compressions and Schur complements, matrix sums and products. In particular, a majorization relation is established between the phases of the nonzero eigenvalues of and the phases of the compressions of and , which leads to a generalized matrix small phase theorem. For the matrices which are not necessarily semi-sectorial, we define their (largest and smallest) essential phases via diagonal similarity transformation. An explicit expression for the essential phases of a Laplacian matrix of a directed graph is obtained.
Keywords
Cite
@article{arxiv.2205.07607,
title = {On the Phases of a Semi-Sectorial Matrix},
author = {Li Qiu and Dan Wang and Xin Mao and Wei Chen},
journal= {arXiv preprint arXiv:2205.07607},
year = {2022}
}
Comments
arXiv admin note: text overlap with arXiv:2105.03630