On the Phases of a Semi-Sectorial Matrix

TL;DR

This paper generalizes the concept of phases from sectorial matrices to semi-sectorial matrices, including singular cases, and establishes new properties and relations, such as a generalized small phase theorem and explicit phases for Laplacian matrices.

Contribution

It extends phase definitions and properties to semi-sectorial matrices, including singular cases, and introduces a generalized small phase theorem and explicit phase formulas for Laplacian matrices.

Findings

Established a majorization relation for phases of eigenvalues of AB

Extended phase properties to Moore-Penrose generalized inverse, compressions, and Schur complements

Derived explicit expressions for the essential phases of Laplacian matrices

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 $AB$ and the phases of the compressions of $A$ and $B$, 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.