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arxiv: 2507.06076 · v2 · submitted 2025-07-08 · 🧮 math.CA · math.CO

Entrywise transforms preserving matrix positivity and non-positivity

Dominique Guillot , Himanshu Gupta , Prateek Kumar Vishwakarma , Chi Hoi Yip This is my paper

Pith reviewed 2026-05-19 05:53 UTC · model grok-4.3

classification 🧮 math.CA math.CO
keywords sign preserversentrywise transformspositive definite matricesfield automorphismsfixed dimensionmatrix positivitynegativity preservationmonotone maps
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The pith

Entrywise functions preserve matrix positive definiteness precisely when they are positive multiples of continuous field automorphisms for fixed sizes of three or larger.

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

The authors characterize functions that preserve the definiteness sign of square matrices when applied entrywise. For any fixed matrix dimension of at least three, such sign preservers turn out to be exactly the positive scalar multiples of the continuous automorphisms of the real or complex numbers. This classification is obtained by first solving the two-by-two case over rich complex domains and then lifting the result to higher dimensions. The work also links to negativity preservation and covers matrices with zero patterns along with monotone maps.

Core claim

When applied entrywise, a function f turns a positive definite matrix into another positive definite matrix, and likewise for negative definite matrices, if and only if f is a positive scalar multiple of a continuous field automorphism, provided the matrix size is fixed and at least three. For two-by-two matrices the preservers instead take the form of extensions of power functions. These conclusions rest on a prior classification of two-by-two entrywise positivity preservers defined over broader complex domains.

What carries the argument

The lifting from the two-by-two classification of entrywise positivity preservers over complex domains to higher fixed dimensions via domain richness that rules out other candidates.

Load-bearing premise

The functions under consideration are defined on sufficiently rich complex domains that allow the two-by-two classification to be lifted to higher dimensions.

What would settle it

A concrete function that preserves both positivity and non-positivity under entrywise application for all three-by-three positive and negative definite matrices but fails to be a positive multiple of a continuous field automorphism would disprove the classification.

read the original abstract

We characterize real and complex functions which, when applied entrywise to square matrices, yield a positive definite matrix if and only if the original matrix is positive definite. We refer to these transformations as sign preservers. Compared to classical work on entrywise preservers of Schoenberg and others, we completely resolve this problem in the harder fixed dimensional setting, extending a similar recent classification of sign preservers obtained for matrices over finite fields. When the matrix dimension is fixed and at least $3$, we show that the sign preservers are precisely the positive scalar multiples of the continuous automorphisms of the underlying field. This is in contrast to the $2 \times 2$ case where the sign preservers are extensions of power functions. These results are built on our classification of $2 \times 2$ entrywise positivity preservers over broader complex domains. Our results yield a complementary connection with a work of Belton, Guillot, Khare, and Putinar (2023) on negativity-preserving transforms. We also extend our sign preserver results to matrices with a structure of zeros, as studied by Guillot, Khare, and Rajaratnam for the entrywise positivity preserver problem. Finally, in the spirit of sign preservers, we address a natural extension to monotone maps, classically studied by Loewner and many others.

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

2 major / 2 minor

Summary. The manuscript characterizes real and complex functions f such that the entrywise application f[A] yields a positive definite matrix if and only if A is positive definite. For fixed dimension n ≥ 3 these sign preservers are precisely the positive scalar multiples of the continuous automorphisms of the underlying field (ℝ or ℂ). For the 2 × 2 case the preservers are extensions of power functions. The classification rests on a preliminary 2 × 2 result over broader complex domains and is extended to matrices with prescribed zero patterns as well as to monotone maps.

Significance. If the central claims hold, the work supplies a complete fixed-dimension classification that complements Schoenberg’s classical variable-dimension results and the recent finite-field classification. The explicit identification with field automorphisms for n ≥ 3 is a clean, falsifiable statement, and the links to negativity preservers and structured matrices add complementary value.

major comments (2)
  1. [Abstract and 2×2 classification] Abstract and the 2 × 2 classification section: the domain on which the 2 × 2 positivity preservers are classified is described only as “broader complex domains.” The lifting argument to n ≥ 3 (used to exclude non-automorphism candidates) requires the domain to contain an open set intersecting the imaginary axis or to be sufficiently dense in ℂ; without an explicit statement of these domain properties the exclusion step cannot be verified.
  2. [Section on lifting to n≥3] Lifting argument for n ≥ 3: the proof that only positive multiples of continuous field automorphisms survive relies on the richness of the 2 × 2 domain to rule out other candidates that might preserve definiteness sign only for the fixed n. If the domain lacks non-real points or is not dense, the argument fails to exclude extra preservers; a concrete lemma verifying the required domain properties is needed.
minor comments (2)
  1. [Introduction] Introduction: the reference to Belton, Guillot, Khare, and Putinar (2023) should appear with full bibliographic details at first citation.
  2. [Notation] Notation section: the entrywise application notation f[A] should be defined explicitly before its first use to distinguish it from ordinary functional calculus.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater precision regarding the domains in the 2×2 classification and the lifting argument. We have revised the manuscript to make these properties explicit and to add a supporting lemma.

read point-by-point responses

  1. Referee: [Abstract and 2×2 classification] Abstract and the 2 × 2 classification section: the domain on which the 2 × 2 positivity preservers are classified is described only as “broader complex domains.” The lifting argument to n ≥ 3 (used to exclude non-automorphism candidates) requires the domain to contain an open set intersecting the imaginary axis or to be sufficiently dense in ℂ; without an explicit statement of these domain properties the exclusion step cannot be verified.

    Authors: We agree that the phrase “broader complex domains” is too vague for the lifting argument to be verified. In the revised version we replace it with an explicit definition: the domains are open subsets of ℂ that either intersect the imaginary axis in an open set or are dense in ℂ. We have also inserted a short paragraph immediately after the definition that records the precise topological conditions used in the subsequent proofs. revision: yes

  2. Referee: [Section on lifting to n≥3] Lifting argument for n ≥ 3: the proof that only positive multiples of continuous field automorphisms survive relies on the richness of the 2 × 2 domain to rule out other candidates that might preserve definiteness sign only for the fixed n. If the domain lacks non-real points or is not dense, the argument fails to exclude extra preservers; a concrete lemma verifying the required domain properties is needed.

    Authors: We accept the referee’s observation. The revised manuscript now contains a new lemma (Lemma 3.4) that states and proves the required domain properties: every domain under consideration contains a nonempty open set intersecting the imaginary axis and is dense in ℂ. The lifting argument then cites this lemma directly, making the exclusion of non-automorphism candidates fully verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the classification derivation

full rationale

The paper derives its main classification for fixed n≥3 by first establishing the 2×2 entrywise positivity preservers over broader complex domains and then lifting via domain richness to exclude non-automorphism candidates. This constitutes a standard forward mathematical argument rather than any reduction of the claimed result to its own inputs by construction. No self-citation (including the complementary 2023 connection or finite-field extension) is load-bearing for the core theorem; the proof chain relies on explicit domain assumptions and case analysis that remain independent of the final statement. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The classification rests on standard properties of positive definite matrices and field automorphisms; no new free parameters or invented entities are introduced. The main background assumptions are continuity of the automorphisms and sufficient richness of the function domain.

axioms (1)
  • domain assumption Continuous automorphisms of the real or complex field are the only candidates that survive the sign-preserver condition for dimension >=3.
    Invoked in the statement of the main theorem for fixed dimension >=3.

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Forward citations

Cited by 1 Pith paper

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

  1. 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

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