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

Charles Johnson; Susana Furtado

arxiv: 2605.16267 · v1 · pith:7YJBG7OJnew · submitted 2026-03-30 · 🧮 math.CO · math.OC

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

Susana Furtado , Charles Johnson This is my paper

Pith reviewed 2026-05-21 09:24 UTC · model grok-4.3

classification 🧮 math.CO math.OC

keywords reciprocal matrixPerron eigenvectororderingexistenceconstruction

0 comments

The pith

Reciprocal matrices can be constructed to have any prescribed orders on the right Perron eigenvector and the entrywise inverse of the left Perron eigenvector.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a general procedure for building reciprocal matrices of size n so that the right Perron eigenvector follows one chosen ordering of its components while the entrywise inverse of the left Perron eigenvector follows a second chosen ordering. The construction works by selecting positive matrix entries that satisfy a collection of inequalities encoding those orderings together with the reciprocity requirement that each off-diagonal pair multiplies to one. An explicit four-by-four matrix is supplied that achieves reverse orderings, settling the existence question for that particular case. If the procedure extends to all sizes and all order pairs, then the two eigenvectors can have their component rankings chosen independently inside the class of reciprocal matrices.

Core claim

The paper establishes that for any pair of specified orders on the right Perron eigenvector and the entrywise inverse of the left Perron eigenvector there exists a positive reciprocal matrix realizing both orderings. A constructive procedure is given that produces such a matrix for arbitrary n, and a concrete four-by-four example is exhibited that realizes the reverse ordering pair.

What carries the argument

The construction procedure that selects positive entries satisfying ordering inequalities while enforcing the reciprocity condition a sub ij times a sub ji equals one.

If this is right

Reciprocal matrices realizing any chosen pair of orderings on the two vectors exist for every size n.
The reverse ordering case is achievable when the matrix has size four.
The orderings of the right and inverse-left Perron vectors can be prescribed independently.
The construction produces an explicit positive reciprocal matrix for each such pair of orders.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same approach may extend to other eigenvector ordering problems or to matrices with additional sign patterns.
Computational checks for small n could map out which order pairs require the fewest free parameters.
Such matrices could serve as test cases for studying how eigenvector rankings interact with consistency measures.

Load-bearing premise

The systems of equations and inequalities that enforce the desired orderings always admit positive real solutions that keep the matrix reciprocal.

What would settle it

A concrete pair of orders for which the associated system of equations has no positive reciprocal solution would disprove the general existence result.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They supply an explicit construction for reciprocal matrices with arbitrary prescribed orders on the right Perron vector and inverse left Perron vector, plus a concrete 4x4 example that settles the Bozoki-Csato question.

read the letter

The paper gives a procedure to build a reciprocal matrix so that the right Perron eigenvector and the entrywise inverse of the left one have any chosen component orderings. They also hand over a size-4 matrix that realizes the reverse ordering, which directly answers the existence question left open in the 2026 Bozoki-Csato manuscript. That is the useful part: a concrete affirmative answer plus a method that appears to work at least for small n. The construction is presented as solving a system of equations and inequalities that enforce the desired orderings while keeping the matrix reciprocal and positive. For the n=4 case they exhibit the actual entries, so that part can be checked directly. The approach does not seem to recycle older parametrizations in the literature they cite, which is a modest plus for the subfield. The main limitation is that the general claim—for every n and every pair of orderings—rests on the assertion that the resulting polynomial systems always admit positive solutions. The manuscript supplies the procedure and the small example but does not include an inductive argument, degree-theoretic guarantee, or explicit parametrization that would confirm feasibility across all cases. Without that step the procedure could stall for some combinations of orders or larger n, even if the n=4 reverse-order instance succeeds. The citation pattern is narrow and focused on the immediate predecessors, which is reasonable for a short existence note but leaves the broader context thin. This work is aimed at people who already care about Perron vectors in reciprocal matrices for decision models or pairwise comparison problems. A reader inside that niche will find the explicit 4x4 counter-example and the construction procedure worth seeing. Outside the niche it is unlikely to matter much. The paper is coherent on its own terms and shows honest engagement with the cited open question, so it clears the bar for a serious referee even though the general existence argument needs tightening. I would send it out for review rather than desk-reject it.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a procedure to construct a reciprocal matrix (with a_ji = 1/a_ij) such that the right Perron eigenvector x and the entrywise inverse of the left Perron eigenvector 1/y realize any prescribed pair of orders. It supplies an explicit 4x4 example realizing reverse orders and thereby answers affirmatively the existence question posed by Bozoki and Csato (2026) for size 4.

Significance. If the general procedure is valid, the result would establish that all combinations of orderings are attainable for the right and inverse-left Perron vectors in reciprocal matrices of any size. This would be a useful clarification in the theory of positive matrices and their spectral properties, with possible relevance to consistency analysis in pairwise-comparison methods.

major comments (1)

[Construction procedure] The central claim asserts a general construction procedure that produces positive reciprocal entries satisfying the strict ordering inequalities on x and 1/y for arbitrary prescribed orders and any n. No existence argument (e.g., inductive construction, topological degree, or explicit parametrization guaranteeing positivity) is supplied for the resulting system of polynomial equations and inequalities when n > 4 or for arbitrary order pairs. The n=4 example demonstrates feasibility in one instance but does not establish the general case.

minor comments (1)

[Abstract] The abstract states the procedure works for 'any pair of given orders' while the body supplies only the size-4 case; a brief clarifying sentence on the scope of the general claim would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for acknowledging the potential significance of the results. We address the major comment below.

read point-by-point responses

Referee: The central claim asserts a general construction procedure that produces positive reciprocal entries satisfying the strict ordering inequalities on x and 1/y for arbitrary prescribed orders and any n. No existence argument (e.g., inductive construction, topological degree, or explicit parametrization guaranteeing positivity) is supplied for the resulting system of polynomial equations and inequalities when n > 4 or for arbitrary order pairs. The n=4 example demonstrates feasibility in one instance but does not establish the general case.

Authors: We appreciate the referee's observation. The manuscript outlines a general procedure for constructing reciprocal matrices with prescribed orders on the right Perron vector and entrywise inverse left Perron vector. However, we agree that a complete, self-contained existence argument (such as an explicit parametrization or inductive construction ensuring positivity of all entries and satisfaction of the strict inequalities for arbitrary n and order pairs) is not fully developed beyond the explicit n=4 case. In the revised manuscript we will add a detailed construction that reduces the problem to choosing positive parameters satisfying a system of strict inequalities, together with a proof that such parameters always exist. revision: yes

Circularity Check

0 steps flagged

No circularity: direct construction with explicit example

full rationale

The paper presents an explicit construction procedure for reciprocal matrices realizing arbitrary prescribed orders on the right Perron vector and entrywise inverse left Perron vector, together with a concrete positive reciprocal matrix of size 4 realizing reverse orders. This construction is given directly via systems of equations and inequalities enforcing reciprocity and the desired component orderings; no step reduces by definition to the target result, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content is unverified. The affirmative answer to the Bozoki-Csato question for n=4 rests on the supplied example rather than on any self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard Perron-Frobenius theory for positive matrices and the algebraic freedom to choose entries that satisfy reciprocity and ordering constraints. No free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Perron-Frobenius theorem guarantees unique positive right and left eigenvectors for irreducible nonnegative matrices.
Invoked implicitly to guarantee the eigenvectors exist and are positive.

pith-pipeline@v0.9.0 · 5603 in / 1148 out tokens · 27293 ms · 2026-05-21T09:24:04.721698+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Boz´ oki, L

S. Boz´ oki, L. Csat´ o,Theoretical properties of the eigenvector method, arXiv:2603.24274 [math.OC]

work page arXiv
[2]

Boz´ oki,Inefficient weights from pairwise comparison matrices with arbi- trarily small inconsistency, Optimization 63 (2014), 1893-1901

S. Boz´ oki,Inefficient weights from pairwise comparison matrices with arbi- trarily small inconsistency, Optimization 63 (2014), 1893-1901

work page 2014
[3]

Csat´ o,Right-left asymmetry of the eigenvector method: A simulation study, European Journal of Operational Research 313 (2024), 708-717

L. Csat´ o,Right-left asymmetry of the eigenvector method: A simulation study, European Journal of Operational Research 313 (2024), 708-717

work page 2024
[4]

C. R. Johnson, W. B. Beine, T. J. Wang,Right-left asymmetry in an eigen- vector ranking procedure, Journal of Mathematical Psychology 19 (1979), 61-64

work page 1979
[5]

Furtado, C

S. Furtado, C. R. Johnson,Efficiency analysis for the Perron vector of a re- ciprocal matrix, Applied Mathematics and Computation 480 (2024), 128913

work page 2024
[6]

T. L. Saaty,A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology 15 (1977), 234-281. 3

work page 1977

[1] [1]

Boz´ oki, L

S. Boz´ oki, L. Csat´ o,Theoretical properties of the eigenvector method, arXiv:2603.24274 [math.OC]

work page arXiv

[2] [2]

Boz´ oki,Inefficient weights from pairwise comparison matrices with arbi- trarily small inconsistency, Optimization 63 (2014), 1893-1901

S. Boz´ oki,Inefficient weights from pairwise comparison matrices with arbi- trarily small inconsistency, Optimization 63 (2014), 1893-1901

work page 2014

[3] [3]

Csat´ o,Right-left asymmetry of the eigenvector method: A simulation study, European Journal of Operational Research 313 (2024), 708-717

L. Csat´ o,Right-left asymmetry of the eigenvector method: A simulation study, European Journal of Operational Research 313 (2024), 708-717

work page 2024

[4] [4]

C. R. Johnson, W. B. Beine, T. J. Wang,Right-left asymmetry in an eigen- vector ranking procedure, Journal of Mathematical Psychology 19 (1979), 61-64

work page 1979

[5] [5]

Furtado, C

S. Furtado, C. R. Johnson,Efficiency analysis for the Perron vector of a re- ciprocal matrix, Applied Mathematics and Computation 480 (2024), 128913

work page 2024

[6] [6]

T. L. Saaty,A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology 15 (1977), 234-281. 3

work page 1977