On a conjecture concerning the extensions of a reciprocal matrix

TL;DR

This paper investigates the properties of reciprocal matrices and their extensions, proving a conjecture about the structure of associated digraphs and analyzing conditions for the efficiency of Perron eigenvectors in perturbed matrices.

Contribution

It proves a conjecture regarding the absence of sources in digraphs of matrix extensions and characterizes when perturbed reciprocal matrices have efficient Perron eigenvectors.

Findings

No extension of a reciprocal matrix has a digraph with a source, confirming the conjecture.

Conditions on matrix entries ensure the Perron eigenvector's efficiency.

Analysis applies to matrices perturbed at four entries, with specific submatrix conditions.

Abstract

Let $A$ be a reciprocal matrix of order $n$ and $w$ be its Perron eigenvector. To infer the efficiency of $w$ for $A$, based on the principle of Pareto optimal decisions, we study the strong connectivity of a certain digraph associated with $A$ and $w$. A reciprocal matrix $B$ of order $n+1$ is an extension of $A$ if the matrix $A$ is obtained from $B$ by removing its last row and column. We prove that there is no extension of a reciprocal matrix whose digraph associated with the extension and its Perron eigenvector has a source, as conjectured by Furtado and Johnson in ``Efficiency analysis for the Perron vector of a reciprocal matrix". As an application, considering $n\geq 5$ and $A$ a matrix obtained from a consistent one by perturbing four entries above the main diagonal, $x,y,z,a$, and the corresponding reciprocal entries, in a way that there is a submatrix of size $2$ containing the four perturbed entries and not containing a diagonal entry, we describe the relations among $x,y,z,a$ with which $A$ always has efficient Perron eigenvector.