Existence of reciprocal matrices with specified orders for the right and inverse left Perron eigenvectors

TL;DR

This paper presents a method to construct reciprocal matrices with specified eigenvector orderings, providing explicit examples and answering a recent open question about eigenvector order reversals.

Contribution

It introduces a construction procedure for reciprocal matrices with prescribed eigenvector orderings and confirms the existence of such matrices of size 4.

Findings

Constructed reciprocal matrices with any pair of eigenvector orders.

Provided explicit example for size 4 matrices.

Confirmed existence of matrices with reversed eigenvector orders.

Abstract

Here we give a procedure to construct a reciprocal matrix for which the right and entrywise inverse left Perron eigenvectors have any pair of given orders. An explicit example when the matrix is of size 4 is presented. In particular, it gives an afirmative answer to the question posed in a recent manuscript by Boz\'oki and Csat\'o (2026) about the existence of a reciprocal matrix of size 4 such that the right and entrywise inverse left Perron eigenvectors have reverse orders.