Cholesky decomposition for symmetric matrices over finite fields

Prateek Kumar Vishwakarma

REVIEW 3 major objections 3 minor 1 cited by

Cholesky decomposition for symmetric matrices over finite fields

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Over finite fields, the LPM cone — symmetric matrices with all leading principal minors nonzero — has density 1 and a Cholesky factorization compatible with Frobenius, group operations, and enumeration.

desk verdict Abstract-only paper with a plausible Cholesky program over finite fields, but the density-1 claim is either true for q→∞ or false for n→∞, and the abstract never says which. read the letter →

arxiv 2508.04657 v2 pith:O7Y2JP5Y submitted 2025-08-06 math.CO math.RA

Prateek Kumar Vishwakarma This is my paper · ORCID

classification math.COmath.RA MSC 15A2305A15

keywords CholeskyfactorizationfinitefieldsleadingprincipalminorsLPMconesymmetricmatricesFrobeniusmapmatrixenumerationgroupoperationson

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper brings the Cholesky factorization, a standard tool for real and complex symmetric matrices, to finite fields. It proves that every symmetric matrix over a finite field whose leading principal minors are all nonzero — the so-called LPM cone — can be written in Cholesky form $A = LDL^T$, and that this cone has asymptotic density 1 as the field grows. The factorization respects the entrywise Frobenius map, supports group operations on the cone, and yields exact enumeration formulas for sub-cones. A sympathetic reader would care because it gives finite-field analogues of a factorization that underlies numerical linear algebra, now with exact counting and algebraic structure.

What carries the argument

The carrying mechanism is the LPM cone: the set of symmetric matrices over $\mathbb{F}_q$ whose leading principal minors are all nonzero. The recursive $LDL^T$ factorization follows from these minors being nonzero, and over a finite field the factorization gives a bijection between the cone and the Cartesian product of nonzero diagonal scalars and strictly lower-triangular entries. This bijection is what makes the density, the Frobenius compatibility, the group operations, and the enumeration all tractable.

What would settle it

Fix $q=2$ and compute the fraction of symmetric $n\times n$ matrices over $\mathbb{F}_2$ with all leading principal minors nonzero. That fraction equals $2^{-n}$, falling from 1/2 at $n=1$ to about 0.001 at $n=10$, so if the paper's density-1 claim is read with $n\to\infty$ for fixed $q$, it is false; the claim is only viable with $q\to\infty$ first.

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Extended reading notes

Core claim

The paper's central claim is that the LPM cone over a finite field — the set of $n\times n$ symmetric matrices with all leading principal minors nonzero — admits a general Cholesky factorization $A = LDL^T$ with $L$ unit lower triangular and $D$ diagonal, and that this cone has asymptotic density $1$ in the regime where $q\to\infty$ with $n$ fixed. The factorization is compatible with the entrywise Frobenius map, meaning the Frobenius image of a matrix factors as the Frobenius image of its Cholesky factors. This compatibility is then used to define group operations on the cone and to enumerate sub-cones by counting the triangular factors.

Load-bearing premise

The density-1 statement holds only in the asymptotic order where the field size $q$ tends to infinity while the matrix dimension $n$ stays fixed; for fixed $q$ and growing $n$, the fraction of symmetric matrices with all leading principal minors nonzero is $\left(\frac{q-1}{q}\right)^n$, which goes to 0.

Editorial extensions

If this is right

As $q$ grows with $n$ fixed, the proportion of $n\times n$ symmetric matrices admitting a Cholesky factorization tends to 1, so the factorization is a near-universal normal form in the large-field regime.
Because the factorization is Frobenius-compatible, entrywise power maps on the cone can be studied through their action on triangular factors, connecting to positivity preservation over finite fields.
The Cholesky coordinates turn the LPM cone into a set with explicit group operations, yielding new finite algebraic structures whose orders are computable from the factorization.
Sub-cones of LPM matrices can be enumerated exactly by counting allowed diagonal and strictly lower-triangular entries, giving closed-form counts for natural subfamilies.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The unstated asymptotic regime matters: if one instead fixes $q$ and lets $n$ grow, the density is $\left(\frac{q-1}{q}\right)^n$ and shrinks to 0; the paper's density-1 claim therefore rests on the $q\to\infty$, $n$ fixed ordering.
The same factorization bijection may give a finite-field analogue of the Cholesky-based parameterization of the positive definite cone, potentially relevant to optimization or coding over finite fields where 'positive definiteness' is otherwise unavailable.
One testable extension would be to classify which sub-cones correspond to prescribed sign patterns of the leading principal minors over $\mathbb{F}_q$, and whether their counts factor into $q$-binomial-like expressions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is an abstract-only submission with a plausible program but a load-bearing ambiguity in the main density claim. The author says the LPM cone over finite fields has asymptotic density 1, but never states the limit. The exact count is q^{n(n−1)/2}(q−1)^n symmetric matrices with all leading principal minors nonzero out of q^{n(n+1)/2} total, so the density is ((q−1)/q)^n. For fixed q and n→∞ that goes to 0, not 1; for q→∞ with n fixed it goes to 1. That's not a minor nitpick—it determines whether the headline is true or false.

What's actually new: transplanting the Khare–Vishwakarma Cholesky factorization from real/complex Hermitian matrices to finite fields, checking compatibility with the entrywise Frobenius map, and using the factorization to define group operations and enumerate sub-cones. That is a natural, coherent extension, and the citations to Cooper–Hanna–Whitlatch and to the author's own prior work are appropriate. I can't check the proofs from the abstract, but there's no obvious reason the program can't be carried out.

Soft spots: besides the density regime, the abstract doesn't say how much of this overlaps with the 2025 Khare–Vishwakarma paper; 'extend' could mean anything from a corollary to a new theorem. The enumeration and group operations may be straightforward once you have the factorization, so the contribution hinges on whether the finite-field Cholesky factorization is as clean as claimed.

My recommendation: if the author clarifies that the intended limit is q→∞ (or n=o(q)), the density-1 line is fine and the paper is probably a solid contribution to finite geometry. If the intended limit is n→∞ for fixed q, the headline is wrong and the whole framing collapses. As an editor, I'd send the full version to a referee—the math is checkable and the idea is worth the referee's time—but I'd ask the author to fix the abstract and state the limit explicitly before acceptance. For your own reading group, it's a maybe: useful if someone cares about finite-field matrix factorizations, but not essential until the density question is settled.

Referee Report

3 major / 3 minor

Summary. The paper (arXiv:2508.04657) proposes a finite-field analogue of the Cholesky decomposition for symmetric matrices whose leading principal minors (LPMs) are nonzero. The abstract announces four contributions: (i) a general Cholesky factorization on the LPM cone over finite fields, with the cone having asymptotic density 1; (ii) compatibility of this factorization with the entrywise Frobenius map; (iii) group operations on the cone defined via Cholesky structures; and (iv) enumeration of sub-cones of LPM matrices using the factorization. The work builds on prior results by Cooper–Hanna–Whitlatch (2024) and Khare–Vishwakarma (2025), and on Frobenius-map positivity preservers by Guillot–Gupta–Vishwakarma–Yip (2025).

Significance. If the claims are correct, the paper would provide a clean structural description of a large subset of symmetric matrices over finite fields, with applications to enumeration and to positivity preservation under Frobenius maps. The explicit counting claim attached to the LPM cone is falsifiable and, if the asymptotic regime is stated precisely, could be verified by elementary arguments. The use of known results rather than circular reasoning is appropriate. However, because the review is based only on the abstract, no proof details are available, and the central density claim is ambiguous as stated, so the significance is conditional.

major comments (3)

[Abstract, item (i)] The claim that the LPM cone has asymptotic density 1 is load-bearing but the asymptotic regime is not specified. For n×n symmetric matrices over F_q, an LDL^T decomposition gives a bijection between matrices with all LPMs nonzero and pairs (L,D) with L unit lower triangular and D diagonal with nonzero entries. Hence the exact density is q^{n(n-1)/2}(q-1)^n / q^{n(n+1)/2} = ((q-1)/q)^n. If the intended regime is n→∞ with q fixed, this density tends to 0, contradicting the abstract. If the intended regime is q→∞ with n fixed, the density does tend to 1, but this must be stated. If both n and q grow, density tends to 1 only when n=o(q). This ambiguity also affects items (iii) and (iv), which inherit the regime dependence. The abstract needs an explicit definition of asymptotic density and the intended limiting process.
[Abstract, items (iii)-(iv)] The group operations and enumeration results are announced without enough detail to assess their validity. It is unclear how 'Cholesky-structures' define a group operation on the LPM cone; ordinary matrix multiplication is not generally closed on matrices with all LPMs nonzero. The enumeration claim in item (iv) presumably follows from the bijection with (L,D) pairs, but the abstract does not state whether uniqueness of the factorization is asserted over finite fields, nor how sub-cones are defined. Without formal definitions and theorem statements, these claims cannot be checked.
[Abstract, item (ii)] The compatibility with the entrywise Frobenius map needs the base field to be specified. If the matrices are over F_q, the Frobenius map x↦x^q is the identity on entries, making the compatibility statement trivial. If the intended setting is extension fields F_{q^m} or Hermitian matrices over complex/real fields, the nontriviality and the exact meaning of 'positive matrices over finite fields' must be clarified. As written, the statement is too vague to constitute a checkable mathematical claim.

minor comments (3)

[Abstract] The abbreviation LPM is used without definition; it is later spelled out in parentheses, but only after the term appears in item (i). Define at first use.
[Abstract, item (i)] The phrase 'dense sub-family' is informal. In a finite field there is no natural topological density; the intended meaning should be stated in terms of the ratio of counts.
[References] The abstract cites Khare–Vishwakarma (2025) and Guillot–Gupta–Vishwakarma–Yip (2025), but the specific results used are not identified. A sentence indicating which theorems are extended would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity in the abstract; the central claim rests on an independent factorization and counting argument.

full rationale

The abstract's claims (i)-(iv) are stated as extensions or applications of known results. The LPM cone over finite fields is defined by nonzero leading principal minors, and the Cholesky/LDL^T factorization is a standard bijection with unit lower-triangular L and nonzero diagonal D. The density claim is a consequence of that count, not a fitted parameter or a definition of the conclusion. The citations to Khare-Vishwakarma (2025) and Guillot-Gupta-Vishwakarma-Yip (2025) are to the authors' own prior work, but they are background results with independent content; the abstract does not invoke a uniqueness theorem or smuggle in an ansatz via those citations. The unstated asymptotic regime (e.g., q→∞ with n fixed) is a correctness/ambiguity concern, not circularity: the exact density ((q−1)/q)^n does not by itself define the target result. Without full text, no equation-level reduction is visible, so no circular step is identified.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities appear in the abstract. The claims rest on importing the real/complex Cholesky machinery to finite fields and on an unspecified asymptotic regime; these are the main assumptions beyond standard finite-field linear algebra.

assumptions (3)

domain assumption The finite-field analogue of Cholesky factorization is A = L D L^T with L unit lower triangular and D diagonal, so that nonvanishing leading principal minors determine the factorization and its count.
The abstract refers to an LPM cone and asymptotic density 1; counting the cone requires a specific decomposition form. The full text must state this theorem.
domain assumption Asymptotic density is taken in the regime q→∞ with n fixed, or some explicitly stated equivalent; the abstract does not state this.
For fixed q and n→∞, the density of matrices with all leading principal minors nonzero is ((q-1)/q)^n, which tends to 0, so a density-1 claim requires the opposite regime.
domain assumption The compatibility with the entrywise Frobenius map relies on the positivity-preserver results of Guillot, Gupta, Vishwakarma, and Yip, imported as established background.
The abstract states compatibility with a recently studied map; without the full theorem statement, the imported conditions and their restrictions are unknown.

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Cite this review

Pith. "Pith review of Cholesky decomposition for symmetric matrices over finite fields." pith.science (2026). https://pith.science/paper/O7Y2JP5Y

@misc{pith2026250804657,
  author       = {Pith},
  title        = {Pith review of: Cholesky decomposition for symmetric matrices over finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O7Y2JP5Y}},
  note         = {Machine review of arXiv:2508.04657}
}

abstract

Inspired by the seminal work of Andr\'e-Louis Cholesky -- whose contributions remain crucial in broader sciences even after more than a century -- Cooper, Hanna and Whitlatch (2024) developed a theory of positive matrices over finite fields, and Khare and Vishwakarma (2025) described a general Cholesky factorization for a dense sub-family of the cone of Hermitian matrices over real/complex fields, whose leading principal minors (LPM) are nonzero. Building on this, we develop a parallel theory within the finite field setting. Specifically $(i)$ we extend the general Cholesky factorization to the LPM cone over finite fields which has asymptotic density $1$. We show that $(ii)$ this factorization is compatible with the entrywise Frobenius map, recently studied in the context of positivity preservers by Guillot, Gupta, Vishwakarma, and Yip [J. Algebra, 2025]. We also $(iii)$ leverage the Cholesky-structures to define meaningful group operations on the matrix cone, and as an application $(iv)$ enumerate sub-cones of LPM matrices using our general Cholesky factorizations.

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