pith. sign in

arxiv: 2310.18041 · v4 · submitted 2023-10-27 · 🧮 math.CA · math.FA

Negativity-preserving transforms of tuples of symmetric matrices

Alexander Belton , Dominique Guillot , Apoorva Khare , Mihai Putinar This is my paper

Pith reviewed 2026-05-24 06:24 UTC · model grok-4.3

classification 🧮 math.CA math.FA
keywords negativity preserverssymmetric matricesabsolute monotonicityhomothetiesmatrix inertiaentrywise transformsSidon setsmultivariable functions
0
0 comments X

The pith

Negativity-preserving transforms on tuples of symmetric matrices are classified by absolute monotonicity in one variable and homotheties in the others.

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

The paper classifies all transforms that preserve negativity when applied to tuples of symmetric matrices. In the single-variable case these are the absolutely monotone functions. In the multi-variable case a separation occurs, with absolute monotonicity allowed for one variable while the remaining variables are restricted to homotheties. The same classification extends to the complex setting. A reader cares because the result shows greater rigidity for inertia-preserving maps than is known for positive-semidefiniteness preservers.

Core claim

The authors obtain the classification of negativity preservers by combining matrix-analysis techniques with test matrices, Sidon sets, and analytic properties of absolutely monotone functions. They then derive the analogous classification in the multi-variable setting, which exhibits a separation of variables with absolute monotonicity on one side and only homotheties on the other. The complex analogue of the result is also established.

What carries the argument

The separation of variables in the multi-variable classification, in which absolute monotonicity governs one argument while homotheties govern the rest.

If this is right

  • In the single-variable setting every negativity preserver must be an absolutely monotone function.
  • In the multi-variable setting the transform factors into an absolutely monotone function of one matrix variable and a homothety for each of the remaining variables.
  • The identical separation holds for the complex analogue of the result.
  • The classification is obtained by verifying the claimed forms on a sufficient collection of test matrices whose spectra involve Sidon sets.

Where Pith is reading between the lines

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

  • The observed rigidity may constrain which nonlinear maps can be used in models that track matrix inertia without altering it.
  • Similar separation phenomena could be sought for other inertia components such as positive-semidefiniteness or signature preservation.
  • The reliance on Sidon sets suggests that arithmetic-progression-free spectra are key to isolating the functional forms.

Load-bearing premise

Well-chosen test matrices together with Sidon sets and properties of absolutely monotone functions are sufficient to characterize every possible negativity-preserving transform.

What would settle it

An explicit function of several variables that preserves negativity for every tuple of symmetric matrices yet is neither absolutely monotone in one variable nor a homothety in the others would falsify the classification.

read the original abstract

Compared to the entrywise transforms which preserve positive semidefiniteness, those leaving invariant the inertia of symmetric matrices reveal a surprising rigidity. We first obtain the classification of negativity preservers by combining recent advances in matrix analysis with some novel arguments relying on well chosen test matrices, Sidon sets from number theory, and analytic properties of absolutely monotone functions. We continue with the analogous classification in the multi-variable setting, revealing for the first time a striking separation of variables, with absolute monotonicity on one side and only homotheties on the other. We conclude with the complex analogue of this result.

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 / 2 minor

Summary. The manuscript classifies negativity-preserving entrywise transforms on tuples of symmetric matrices. In the single-variable case, it combines recent advances in matrix analysis with test matrices, Sidon sets, and analytic properties of absolutely monotone functions. In the multi-variable setting it establishes a separation of variables, with absolute monotonicity on one side and only homotheties on the other, and concludes with the complex analogue.

Significance. If the classifications hold, the work demonstrates surprising rigidity for negativity preservers relative to positive-semidefiniteness preservers and supplies the first such separation result in the multi-variable case. The integration of matrix-analytic techniques with number-theoretic tools (Sidon sets) and properties of absolutely monotone functions is a methodological strength.

minor comments (2)
  1. [Abstract] Abstract, paragraph 2: the phrase 'striking separation of variables' is used without a forward reference to the precise statement (e.g., Theorem X.Y) that encodes the separation; adding such a pointer would improve readability.
  2. The manuscript invokes 'well-chosen test matrices' and Sidon sets to characterize all transforms; a brief remark on why the chosen families are exhaustive (or a pointer to the relevant lemma) would clarify the completeness argument for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing its significance in demonstrating rigidity of negativity preservers, and for recommending minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation classifies negativity preservers on tuples of symmetric matrices by combining external number-theoretic tools (Sidon sets), analytic characterizations of absolutely monotone functions, and well-chosen test matrices. These are independent of the target result and do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The multi-variable separation into absolute monotonicity versus homotheties follows from the same external machinery rather than any internal renaming or ansatz smuggling. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard properties of absolutely monotone functions and recent matrix-analysis theorems; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Properties of absolutely monotone functions suffice, together with test matrices and Sidon sets, to characterize all negativity preservers.
    Invoked in the classification argument for both single and multi-variable cases.

pith-pipeline@v0.9.0 · 5627 in / 1132 out tokens · 18473 ms · 2026-05-24T06:24:57.658887+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.

  • Cost.FunctionalEquation washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Theorem B ... separation of variables, with absolute monotonicity on one side and only homotheties on the other

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.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Entrywise transforms preserving matrix positivity and non-positivity

    math.CA 2025-07 unverdicted novelty 7.0

    The paper gives a complete classification of entrywise sign preservers of positive definiteness for fixed matrix dimensions over the reals and complexes.

  2. The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings

    math.CA 2025-11 unverdicted novelty 2.0

    A survey of dimension-free entrywise positivity preservers with links to metric embeddings, Schur polynomials, finite fields, and an appendix on sphere packings via Schoenberg's theorem.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 2 Pith papers

  1. [1]

    Atzmon and A

    A. Atzmon and A. Pinkus. Rank restricting functions. Linear Algebra Appl. , 372:305–323, 2003

    work page 2003

  2. [2]

    T. Ya. Azizov and I.S. Iokhvidov. Linear operators in spaces with an indefinite metric . Translated from the Russian by E. R. Dawson. John Wiley & Sons, Ltd., Chic hester, 1989

    work page 1989

  3. [3]

    Belton, D

    A. Belton, D. Guillot, A. Khare, and M. Putinar. A panoram a of positivity. Part II: Fixed dimen- sion. In: Complex Analysis and Spectral Theory , Proceedings of the CRM Workshop held at Laval University, QC, May 21–25, 2018 (G. Dales, D. Khavinson, J. M ashreghi, Eds.). CRM Proceed- ings – AMS Contemporary Mathematics 743, pp. 109–150, Ameri can Mathemat...

    work page 2018

  4. [4]

    Belton, D

    A. Belton, D. Guillot, A. Khare, and M. Putinar. Moment-s equence transforms. J. Eur. Math. Soc., 24(9):3109–3160, 2022

    work page 2022

  5. [5]

    Belton, D

    A. Belton, D. Guillot, A. Khare, and M. Putinar. Matrix co mpression along isogenic blocks. Acta Sci. Math. (Szeged) 88(1-2):417–448, 100th anniversary special issue, 2022

    work page 2022

  6. [6]

    Belton, D

    A. Belton, D. Guillot, A. Khare, and M. Putinar. Totally p ositive kernels, P´ olya frequency func- tions, and their transforms. J. d’Analyse Math. , 150(1):83–158, 2023

    work page 2023

  7. [7]

    Bernstein

    S. Bernstein. Sur les fonctions absolument monotones. Acta Math. , 52:1–66, 1929

    work page 1929

  8. [8]

    Bose and S

    R.C. Bose and S. Chowla. Theorems in the additive theory o f numbers. Comment. Math. Helv. , 37:141–147, 1962

    work page 1962

  9. [9]

    Erd¨ os and P

    P. Erd¨ os and P. Tur´ an. On a problem of Sidon in additive n umber theory, and on some related problems. J. London Math. Soc. , 16(4):212–215, 1941

    work page 1941

  10. [10]

    FitzGerald, C.A

    C.H. FitzGerald, C.A. Micchelli, and A. Pinkus. Functi ons that preserve families of positive semi- definite matrices. Linear Algebra Appl. , 221:83–102, 1995

    work page 1995

  11. [11]

    Guillot, A

    D. Guillot, A. Khare, and B. Rajaratnam. Preserving pos itivity for rank-constrained matrices. Trans. Amer. Math. Soc. , 369(9):6105–6145, 2017

    work page 2017

  12. [12]

    Guillot and B

    D. Guillot and B. Rajaratnam. Retaining positive defini teness in thresholded matrices. Linear Algebra Appl., 436(11):4143–4160, 2012

    work page 2012

  13. [13]

    Henkin and A.A

    G.M. Henkin and A.A. Shananin. Bernstein theorems and R adon transform. Application to the theory of production functions. Translated from the Russia n by S.I. Gelfand. In: Mathematical problems of tomography (I.M. Gelfand and S.G. Gindikin, Eds.), Translations of Mat hematical Monographs 81, pp. 189–223, American Mathematical Society , Providence, 1985

    work page 1985

  14. [14]

    C.S. Herz. Fonctions op´ erant sur les fonctions d´ efini es-positives. Ann. Inst. Fourier (Grenoble) , 13(1):161–180, 1963

    work page 1963

  15. [15]

    R.A. Horn. The theory of infinitely divisible matrices a nd kernels. Trans. Amer. Math. Soc. , 136:269–286, 1969

    work page 1969

  16. [16]

    Horn and C.R

    R.A. Horn and C.R. Johnson. Matrix analysis . Second edition. Cambridge University Press, 2013

    work page 2013

  17. [17]

    Keller-Ressel and S

    M. Keller-Ressel and S. Nargang. The hyperbolic geomet ry of financial networks. Scientific Reports, 11, art. #4732, 12 pp., 2021

    work page 2021

  18. [18]

    A. Khare. Matrix analysis and entrywise positivity preservers . London Math. Soc. Lecture Note Ser. 471, Cambridge University Press, 2022. Also Vol. 82, TR IM Series, Hindustan Book Agency, 2022. 42 ALEXANDER BELTON, DOMINIQUE GUILLOT, APOOR V A KHARE, AND MIHAI PUTINAR

    work page 2022

  19. [19]

    Khrulkov, L

    V. Khrulkov, L. Mirvakhabova, E. Ustinova, I. Oseledet s, and V. Lempitsky. Hyperbolic im- age embeddings. IEEE/CVF Conference on Computer Vision and Pattern Recogni tion (CVPR) , pp. 6417–6427, 2020

    work page 2020

  20. [20]

    Kleinberg

    R. Kleinberg. Geographic routing using hyperbolic spa ce. 26th IEEE International Conference on Computer Communications , pp. 1902–1909, 2007

    work page 1902

  21. [21]

    Pontryagin

    L.S. Pontryagin. Hermitian operators in spaces with in definite metric. Izv. Akad. Nauk SSSR Ser. Mat., 8(6):243–280, 1944

    work page 1944

  22. [22]

    P. Ressel. Higher order monotonic functions of several variables. Positivity, 18(2):257–285, 2014

    work page 2014

  23. [23]

    W. Rudin. Positive definite sequences and absolutely mo notonic functions. Duke Math. J. , 26(4):617–622, 1959

    work page 1959

  24. [24]

    F. Sala, C. De Sa, A. Gu, and C. R´ e. Representation trade offs for hyperbolic embeddings. Proc. Mach. Learn. Res. , 80:4460–4469, 2018

    work page 2018

  25. [25]

    Schoenberg

    I.J. Schoenberg. On finite-rowed systems of linear ineq ualities in infinitely many variables. II. Trans. Amer. Math. Soc. , 35(2):452–478, 1933

    work page 1933

  26. [26]

    Schoenberg

    I.J. Schoenberg. Positive definite functions on sphere s. Duke Math. J. , 9(1):96–108, 1942

    work page 1942

  27. [27]

    I. Schur. Bemerkungen zur Theorie der beschr¨ ankten Bi linearformen mit unendlich vielen Ver¨ anderlichen.J. reine angew. Math. , 140:1–28, 1911

    work page 1911

  28. [28]

    J. Singer. A theorem in finite projective geometry and so me applications to number theory. Trans. Amer. Math. Soc. , 43(3):377–385, 1938

    work page 1938

  29. [29]

    Vasudeva

    H.L. Vasudeva. Positive definite matrices and absolute ly monotonic functions. Indian J. Pure Appl. Math., 10(7):854–858, 1979

    work page 1979

  30. [30]

    Verbeek and S

    K. Verbeek and S. Suri. Metric embedding, hyperbolic sp ace, and social networks. Comput. Geom., 59:1–12, 2016. (A. Belton) School of Engineering, Computing and Mathematics, Univers ity of Ply- mouth, UK Email address : alexander.belton@plymouth.ac.uk (D. Guillot) University of Delaw are, New ark, DE, USA Email address : dguillot@udel.edu (A. Khare) Depar...

    work page 2016