pith. sign in

arxiv: 2605.23552 · v1 · pith:7KB54IRMnew · submitted 2026-05-22 · 🧮 math.CO

The INIEP: Irreducible and Positive Realizations

C. R. Johnson , C. Mariju\'an , M. Pisonero This is my paper

Pith reviewed 2026-05-25 04:18 UTC · model grok-4.3

classification 🧮 math.CO
keywords nonnegative inverse eigenvalue problemirreducible nonnegative matricestrace-zero spectragraph-theoretic characterizationpositive realizationscharacteristic polynomialsNIEPINIEP
0
0 comments X

The pith

Nonnegatively realizable spectra often admit irreducible nonnegative matrix realizations, characterized via graphs when the trace is zero.

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

The paper focuses on the irreducible nonnegative inverse eigenvalue problem: when a list of numbers that can appear as the eigenvalues of some nonnegative matrix can also appear as the eigenvalues of an irreducible nonnegative matrix. Irreducibility means the matrix cannot be rearranged into a block-triangular form with a zero block, corresponding to its directed graph being strongly connected. The authors prove existence of irreducible realizations in several broad classes of spectra. They then give an exact graph-theoretic characterization of which trace-zero realizable polynomials have irreducible realizations. The related question of strictly positive realizations is treated for spectra whose trace is positive.

Core claim

Irreducible nonnegative realizations exist for various classes of spectra already known to be nonnegatively realizable. For dimensions less than five, where the full nonnegative inverse eigenvalue problem is solved, this yields concrete information on irreducible cases. In the trace-zero setting, a graph-theoretic criterion completely decides which realizable polynomials possess an irreducible nonnegative realization. Positive realizations are addressed when the trace is positive.

What carries the argument

Graph-theoretic translation of a characteristic polynomial into conditions on a directed graph that determine whether any realizing nonnegative matrix must be reducible.

If this is right

  • For trace zero, realizability by an irreducible nonnegative matrix is decided exactly by whether the polynomial meets the stated graph conditions.
  • Existence of irreducible realizations holds in the general cases identified without requiring extra checks beyond nonnegative realizability.
  • For every dimension less than five the irreducible versions of all known realizable spectra can be identified explicitly.
  • Spectra with positive trace admit strictly positive matrix realizations under the conditions derived in the paper.

Where Pith is reading between the lines

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

  • The graph criterion supplies a practical test that could be used to construct or rule out strongly connected nonnegative matrices in applications such as population dynamics.
  • The same translation technique may extend to open cases of the NIEP in higher dimensions by suggesting candidate graphs to test.
  • Results on positive realizations suggest a parallel line of inquiry for the positive inverse eigenvalue problem when the trace is strictly positive.

Load-bearing premise

The spectrum is already known to be realizable by some nonnegative matrix, and the graph conditions fully capture the extra requirement of irreducibility with no further algebraic obstructions.

What would settle it

A concrete trace-zero polynomial that is realizable by a nonnegative matrix yet fails to satisfy the graph criterion for irreducibility, or satisfies the criterion yet has no irreducible realization.

read the original abstract

Our focus is upon {\it irreducible} nonnegative $n$-by-$n$ matrix realizations of nonnegatively realizable spectra or, equivalently, characteristic polynomials. After giving some general background, we make some useful new observations and show the existence of irreducible nonnegative realizations in some general cases. Then, we focus on $n<5$, where the NIEP is solved. Finally, we focus on the trace 0 case and, using graph theoretic methods, characterize nonnegative irreducible realizability among realizable polynomials. The closely related problem of positive realizations, for trace positive spectra, is also discussed.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates the irreducible nonnegative inverse eigenvalue problem (INIEP). It provides background on nonnegative realizations, presents new observations and existence results for irreducible nonnegative matrices realizing certain spectra in general cases, specializes to dimensions n<5 (where the NIEP is solved), and for the trace-zero case employs graph-theoretic methods on associated digraphs to characterize which realizable polynomials admit irreducible nonnegative realizations. It also treats the related problem of positive realizations for spectra with positive trace.

Significance. If the characterizations hold, the work strengthens the NIEP literature by systematically incorporating the irreducibility condition, which is required in applications involving strongly connected or primitive nonnegative matrices. The explicit separation of the standard nonnegative realizability precondition from the additional irreducibility analysis, together with the combinatorial translation via digraph connectivity for the trace-zero case, supplies a clean modular framework that can be built upon. The focus on solved low-dimensional cases and the graph-theoretic approach constitute concrete, falsifiable contributions.

minor comments (3)
  1. [§1] §1 (Introduction): the statement that the NIEP is solved for n<5 should include a brief citation to the relevant references (e.g., the known complete solutions) to allow readers to locate the precise lists of realizable spectra.
  2. The graph-theoretic characterization in the trace-zero section relies on connectivity conditions; a short remark clarifying whether the digraph is simple or allows multiple arcs would remove potential ambiguity in the translation from matrix to graph.
  3. Notation for the characteristic polynomial and the associated digraph should be introduced with a single consistent symbol set at the beginning of the trace-zero section to aid cross-referencing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on irreducible nonnegative realizations and for recommending minor revision. We appreciate the recognition of the combinatorial translation via digraphs for the trace-zero case and the overall modular framework.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation separates the precondition of nonnegative realizability (a standard external input from the solved NIEP for n<5) from the graph-theoretic characterization of irreducibility for trace-zero cases. No step reduces a claimed prediction or existence result to a parameter fitted inside the paper, a self-definitional loop, or a load-bearing self-citation chain; the work applies established linear-algebra and digraph connectivity facts to an already-realizable spectrum without re-deriving the realizability condition itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard facts from nonnegative matrix theory and directed-graph interpretations of irreducibility; no free parameters, new entities, or ad-hoc axioms introduced in the abstract.

axioms (2)
  • standard math Standard algebraic properties of characteristic polynomials of nonnegative matrices
    Invoked throughout the background and existence arguments.
  • domain assumption Equivalence between matrix irreducibility and strong connectivity of the associated directed graph
    Central to the trace-zero characterization.

pith-pipeline@v0.9.0 · 5627 in / 971 out tokens · 26116 ms · 2026-05-25T04:18:12.799012+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

21 extracted references · 21 canonical work pages

  1. [1]

    Brauer, Limits for the characteristic roots of a matrix, IV: Applications to stochastic ma- trices,Duke Math

    A. Brauer, Limits for the characteristic roots of a matrix, IV: Applications to stochastic ma- trices,Duke Math. J.19 (1952) 75-91

    work page 1952

  2. [2]

    D. M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, Johann Ambrosius Barth, Heidelberg 1995

    work page 1995

  3. [3]

    Fiedler, Eigenvalues of nonnegative symmetric matrices,Linear Algebra Appl.9 (1974) 119-142

    M. Fiedler, Eigenvalues of nonnegative symmetric matrices,Linear Algebra Appl.9 (1974) 119-142

    work page 1974

  4. [4]

    Frobenius, ¨Uber matrizen aus nicht negativen elementen,S

    G. Frobenius, ¨Uber matrizen aus nicht negativen elementen,S. -B. K. Preuss. Akad. Wiss. Berlin, 1912 p. 456-477. 26

    work page 1912

  5. [5]

    F. R. Gantmacher, Thèorie des matrices, Tome 2, Collection Universitaire de Mathèmatiques, Dunod, Paris, 1966

    work page 1966

  6. [6]

    R. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, 2013

    work page 2013

  7. [7]

    C. R. Johnson, Row stochastic matrices similar to doubly stochastic matrices,Linear Multilin- ear Algebra10 (1981) 113-130

    work page 1981

  8. [8]

    C. R. Johnson, C. Marijuán, P. Papparella, M. Pisonero, The NIEP. Operator theory, opera- tor algebras, and matrix theory, 199-220, Oper. Theory Adv. Appl. 267, Birkh¨ auser/Springer (2018), 353-372. ISBN: 978-3-319-72448-5

    work page 2018

  9. [9]

    C. R. Johnson, P. Paparella, Row cones, perron similarities, and nonnegative spectra,Linear Multilinear Algebra65 (2017) 2124-2130

    work page 2017

  10. [10]

    A. I. Julio, C. Marijuán, M. Pisonero, R. L. Soto, On universal realizability of spectra,Linear Algebra Appl.563 (2019) 353-372

    work page 2019

  11. [11]

    A. I. Julio, C. Marijuán, M. Pisonero, R. L. Soto, Universal realizability in low dimension, Linear Algebra Appl.619 (2021) 107-136

    work page 2021

  12. [12]

    T. J. Laffey, E. Meehan, A characterization of trace zero nonnegative5×5matrices,Linear Algebra Appl.302/303 (1999) 295-302

    work page 1999

  13. [13]

    Loewy, D

    R. Loewy, D. London, A note on the inverse problem for nonnegative matrices,Linear and Multilinear Algebra6 (1978) 83-90

    work page 1978

  14. [14]

    Minc, Inverse elememtary divisor problem for nonnegative matricesProceedings of the Amer- ican Mathematical Society, 83 n4(1981) 665-669

    H. Minc, Inverse elememtary divisor problem for nonnegative matricesProceedings of the Amer- ican Mathematical Society, 83 n4(1981) 665-669

    work page 1981

  15. [15]

    Paparella, Realizing Suleimanova-type spectra via permutative matrices,Electron

    P. Paparella, Realizing Suleimanova-type spectra via permutative matrices,Electron. J. Linear Algebra31 (2016) 306-312

    work page 2016

  16. [16]

    Perfect, On positive stochastic matrices with real characteristic roots,Proceedings of the Cambridge Philosophical Society, 48 (1952) 271-276

    H. Perfect, On positive stochastic matrices with real characteristic roots,Proceedings of the Cambridge Philosophical Society, 48 (1952) 271-276

    work page 1952

  17. [17]

    Perfect, Methods of constructing certain stochastic matrices II,Duke Math.J.22 (1955) 305-311

    H. Perfect, Methods of constructing certain stochastic matrices II,Duke Math.J.22 (1955) 305-311

    work page 1955

  18. [18]

    Perfect, L

    H. Perfect, L. Mirsky, Spectral properties of doubly stochastic matrices,Monatsh. Math.69 (1965) 35-57

    work page 1965

  19. [19]

    Spector, A characterization of trace zero symmetric nonnegative 5×5 matrices,Linear Al- gebra Appl.434 (2011), n 4, 1000-1017

    O. Spector, A characterization of trace zero symmetric nonnegative 5×5 matrices,Linear Al- gebra Appl.434 (2011), n 4, 1000-1017

    work page 2011

  20. [20]

    H. R. Suleˇ ımanova, Stochastic matrices with real characteristic values,Dokl. Akad. Nauk. S.S.S.R.66 (1949) 343-345 (in Russian)

    work page 1949

  21. [21]

    Torre-Mayo, M

    J. Torre-Mayo, M. R. Abril-Raymundo, E. Alarcia-Estévez, C. Marijuán, M. Pisonero, The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs,Linear Algebra Appl.426 (2007) 729-773

    work page 2007