Extended Perron complements of <i>M</i>-matrices

Qin Zhong; Chunyan Zhao; Qin Zhong; Chunyan Zhao

doi:10.3934/math.20231346

AIMS Mathematics

2023, Volume 8, Issue 11: 26372-26383. doi: 10.3934/math.20231346

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Extended Perron complements of M-matrices

Qin Zhong ^,,
Chunyan Zhao

Department of Mathematics, Sichuan University Jinjiang College, Meishan 620860, China

Received: 21 June 2023 Revised: 18 August 2023 Accepted: 01 September 2023 Published: 14 September 2023
MSC : 15A47

This paper aims to consider the extended Perron complements for the collection of M-matrices. We first exhibit the connection between the extended Perron complements of M-matrices and nonnegative matrices. Moreover, we present some common inequalities involving extended Perron complements, Schur complements, and principal submatrices of irreducible M-matrices by utilizing the properties of M-matrices. We also discuss the monotonicity of the extended Perron complements and minimum eigenvalue. For the collection of M-matrices, we demonstrate that all (extended) Perron complements are M-matrices. Especially, we deduce that M-matrices and their Perron complements share the same minimum eigenvalue. Finally, a simple example is presented to illustrate our findings.
- M-matrix,
- Z-matrix,
- minimum eigenvalue,
- extended Perron complement,
- Schur complement
Citation: Qin Zhong, Chunyan Zhao. Extended Perron complements of M-matrices[J]. AIMS Mathematics, 2023, 8(11): 26372-26383. doi: 10.3934/math.20231346

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Abstract

This paper aims to consider the extended Perron complements for the collection of M-matrices. We first exhibit the connection between the extended Perron complements of M-matrices and nonnegative matrices. Moreover, we present some common inequalities involving extended Perron complements, Schur complements, and principal submatrices of irreducible M-matrices by utilizing the properties of M-matrices. We also discuss the monotonicity of the extended Perron complements and minimum eigenvalue. For the collection of M-matrices, we demonstrate that all (extended) Perron complements are M-matrices. Especially, we deduce that M-matrices and their Perron complements share the same minimum eigenvalue. Finally, a simple example is presented to illustrate our findings.

References

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