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.

The concept of the generalized Perron complement concerning a nonnegative irreducible matrix was proposed by L. Z. Lu in 2002, and it was used to construct an algorithm for estimating the boundary of the spectral radius. In this study, we consider the properties of generalized Perron complements of nonnegative irreducible and diagonally dominant matrices. Moreover, we analyze the closure property of the generalized Perron complements of nonnegative irreducible $H$-matrices under certain conditions.

This paper introduced a smoothing algorithm for calculating the maximal eigenvalue of non-defective positive matrices. Two special matrices were constructed to provide monotonically increasing lower-bound estimates and monotonically decreasing upper-bound estimates of the maximal eigenvalue. The monotonicity and convergence of these estimations was also proven. Finally, the effectiveness of the algorithm was demonstrated with numerical examples.

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.