Extended Perron complements of M-matrices

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.

Automated summary

This paper investigates the properties of the extended Perron complements for M matrices. We establish the connection between the extended Perron complements and nonnegative matrices, and present common inequalities involving the extended Perron complements, Schur complements, and principal submatrices of irreducible M matrices by utilizing the properties of M matrices. Furthermore, we discuss the monotonicity of the extended Perron complements and minimum eigenvalue. It is demonstrated that all (extended) Perron complements for the collection of M matrices are also M matrices, and M matrices and their Perron complements share the same minimum eigenvalue. A simple example is provided to illustrate the findings.