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.