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

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 Bozóki and Csató (2026) about the existence of a reciprocal matrix of size 4 such that the right and entrywise inverse left Perron eigenvectors have reverse orders.