Generalized Perron complements of strictly generalized doubly diagonally dominant matrices

Qin Zhong; Ling Li; Gufang Mou; Qin Zhong; Ling Li; Gufang Mou

doi:10.3934/math.2025629

AIMS Mathematics

2025, Volume 10, Issue 6: 13996-14011. doi: 10.3934/math.2025629

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Generalized Perron complements of strictly generalized doubly diagonally dominant matrices

Qin Zhong ^{1
,
,},
Ling Li ^{2
,
,},
Gufang Mou ³

1.
Department of Mathematics, Sichuan University Jinjiang College, Meishan 620860, China
2.
School of Big Data and Artificial Intelligence, Chengdu Technological University, Chengdu 611730, China
3.
College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China

Received: 03 April 2025 Revised: 07 June 2025 Accepted: 13 June 2025 Published: 18 June 2025
MSC : 15A45, 15A48

It is well known that there is an intrinsic connection between Perron complements and Schur complements. It has been demonstrated that Schur complements of strictly generalized doubly diagonally dominant matrices retain the property of strict generalized double diagonal dominance. Our primary aim of this study is to extend these findings to generalized Perron complements of nonnegative irreducible matrices. Specifically, we established that generalized Perron complements derived from strictly generalized doubly diagonally dominant and nonnegative irreducible matrices preserve strict generalized double diagonal dominance and nonnegative irreducibility. Numerical examples are provided to substantiate our theoretical results.
- nonnegative matrix,
- generalized Perron complement,
- diagonally dominant matrix,
- generalized doubly diagonally dominant matrix
Citation: Qin Zhong, Ling Li, Gufang Mou. Generalized Perron complements of strictly generalized doubly diagonally dominant matrices[J]. AIMS Mathematics, 2025, 10(6): 13996-14011. doi: 10.3934/math.2025629

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Abstract

It is well known that there is an intrinsic connection between Perron complements and Schur complements. It has been demonstrated that Schur complements of strictly generalized doubly diagonally dominant matrices retain the property of strict generalized double diagonal dominance. Our primary aim of this study is to extend these findings to generalized Perron complements of nonnegative irreducible matrices. Specifically, we established that generalized Perron complements derived from strictly generalized doubly diagonally dominant and nonnegative irreducible matrices preserve strict generalized double diagonal dominance and nonnegative irreducibility. Numerical examples are provided to substantiate our theoretical results.

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