On the phases of a semi-sectorial matrix and the essential phase of a Laplacian

Wang，Dan1; Mao，Xin2; Chen，Wei3; Qiu，Li2,4

2023-11-01

10.1016/j.laa.2023.07.014

Linear Algebra and Its Applications   Impact Factor & Quartile

0024-3795

1873-1856

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, an interlacing relation is established between the phases of A+B and those of A and B combined. Also, 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.

Essential phase Laplacian Majorization Matrix small phase theorem Phase Semi-sectorial matrix

SCI

English

Others

National Natural Science Foundation of China[62073003];National Natural Science Foundation of China[72131001];

Mathematics

Mathematics, Applied ; Mathematics

WOS:001071954700001

ELSEVIER SCIENCE INC

MATHEMATICS

2-s2.0-85166981483

Scopus

1.School of Electrical Engineering and Computer Science,KTH Royal Institute of Technology,Stockholm,Sweden
2.Department of Electronic and Computer Engineering,Hong Kong University of Science and Technology,Kowloon,Clear Water Bay,Hong Kong
3.Department of Mechanics and Engineering Science,State Key Laboratory for Turbulence and Complex Systems,Peking University,Beijing,100871,China
4.Southern University of Science and Technology,Shenzhen,China

Wang，Dan,Mao，Xin,Chen，Wei,et al. On the phases of a semi-sectorial matrix and the essential phase of a Laplacian[J]. Linear Algebra and Its Applications,2023,676:441-458.

Wang，Dan,Mao，Xin,Chen，Wei,&Qiu，Li.(2023).On the phases of a semi-sectorial matrix and the essential phase of a Laplacian.Linear Algebra and Its Applications,676,441-458.

Wang，Dan,et al."On the phases of a semi-sectorial matrix and the essential phase of a Laplacian".Linear Algebra and Its Applications 676(2023):441-458.