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

Abstract

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