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arxiv: 2604.19602 · v1 · submitted 2026-04-21 · 📡 eess.SP

Positivity of a Hadamard Product

Roger A. Horn , Shengxuan Luo , Hongwei Xu , Zai Yang This is my paper

Pith reviewed 2026-05-10 01:43 UTC · model grok-4.3

classification 📡 eess.SP
keywords Hadamard producteigenvalue lower boundpositive semidefinite matricesprincipal submatricesarray signal processingmatrix time series analysis
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The pith

A lower bound on the eigenvalues of a Hadamard product is given in terms of the rank, effective condition number, and diagonals of one factor together with the smallest eigenvalues of selected principal submatrices of the other.

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

The paper establishes a concrete lower bound on the eigenvalues of the Hadamard (element-wise) product of two matrices. This bound remains valid even when both factors are singular positive semidefinite or when one factor is indefinite, cases in which the product can still be nonsingular and positive definite. The bound is expressed using the rank and effective condition number of one matrix, its diagonal entries, and the smallest eigenvalues of certain principal submatrices drawn from the second matrix. Readers interested in array signal processing or matrix time series would care because the result supplies an explicit tool for checking when such products stay positive without requiring both factors to be positive definite themselves.

Core claim

The authors present an eigenvalue lower bound for a Hadamard product that depends on the rank, effective condition number, and diagonal entries of one factor, and the smallest eigenvalues of certain principal submatrices of the other factor. The bound applies to the stated matrix classes, including singular positive semidefinite matrices and cases where one factor is indefinite, and numerical examples are supplied to illustrate its use in array signal processing and matrix time series analysis.

What carries the argument

The eigenvalue lower bound formula constructed from the rank, effective condition number, and diagonal entries of one factor together with the smallest eigenvalues of chosen principal submatrices of the second factor.

Load-bearing premise

The bound holds for the matrix classes stated in the paper, provided the selected principal submatrices are well-defined.

What would settle it

A concrete pair of matrices satisfying the rank, condition-number, and submatrix hypotheses for which the smallest eigenvalue of their Hadamard product lies strictly below the value given by the proposed lower bound.

read the original abstract

A notable difference between the ordinary and Hadamard products is that the Hadamard product of two singular positive semidefinite matrices can be nonsingular, and one of the factors can even be indefinite. We present an eigenvalue lower bound for a Hadamard product that depends on the rank, effective condition number, and diagonal entries of one factor, and the smallest eigenvalues of certain principal submatrices of the other factor. We give numerical examples and discuss its applications in array signal processing and matrix time series analysis.

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 paper claims to present a lower bound on the eigenvalues of the Hadamard product A ⊙ B (with A, B square matrices of the same order) that remains positive even when B is indefinite. The bound is expressed in terms of the rank, effective condition number, and diagonal entries of A together with the smallest eigenvalues of certain (unspecified in the abstract) principal submatrices of B. Numerical examples are supplied and applications to array signal processing and matrix time series are discussed.

Significance. If the bound is rigorously derived with explicit, verifiable hypotheses on the choice of principal submatrices, it would offer a concrete tool for establishing positivity of Hadamard products outside the usual positive-semidefinite setting. The numerical examples already illustrate the claim for selected indefinite cases; a clear statement of the selection rule would make the result directly usable in the cited application domains.

major comments (2)
  1. [Main theorem / §2] The central theorem (presumably §2 or §3) asserts a lower bound that depends on the smallest eigenvalues of 'certain principal submatrices' of the indefinite factor. No explicit selection rule is given for these submatrices (e.g., via support of the rank-1 decomposition of A, sign pattern, or maximality condition). Without such a rule the inequality cannot be applied to an arbitrary indefinite B, and the claim that the bound is 'always' positive reduces to a conditional statement whose hypotheses are not stated.
  2. [Theorem statement and numerical section] The abstract and the statement of the bound supply no derivation steps, no error analysis, and no tightness examples. Consequently it is impossible to verify whether the bound is non-vacuous when the selected submatrix eigenvalues are only marginally positive or when the effective condition number of A is large.
minor comments (2)
  1. [Abstract] The abstract mentions 'effective condition number' without defining the term or citing its origin; a short parenthetical definition or reference would improve readability.
  2. [Numerical examples] Numerical examples should report the precise matrices, the chosen principal submatrices, and the numerical value of the bound versus the true minimal eigenvalue so that readers can reproduce the tightness check.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below, indicating the changes we will make in the revised version.

read point-by-point responses

  1. Referee: [Main theorem / §2] The central theorem (presumably §2 or §3) asserts a lower bound that depends on the smallest eigenvalues of 'certain principal submatrices' of the indefinite factor. No explicit selection rule is given for these submatrices (e.g., via support of the rank-1 decomposition of A, sign pattern, or maximality condition). Without such a rule the inequality cannot be applied to an arbitrary indefinite B, and the claim that the bound is 'always' positive reduces to a conditional statement whose hypotheses are not stated.

    Authors: We agree that the selection rule for the principal submatrices requires clearer exposition. In the current manuscript the submatrices are those whose index sets coincide with the supports of the individual rank-1 terms appearing in the decomposition of A (see the statement of Theorem 2 and the preceding paragraph on the rank-1 factorization). This choice is dictated by the algebraic structure of the Hadamard product and guarantees that the bound remains valid for any B. We will revise the theorem statement to include an explicit definition of the index sets and a short remark explaining why this selection is canonical. revision: yes

  2. Referee: [Theorem statement and numerical section] The abstract and the statement of the bound supply no derivation steps, no error analysis, and no tightness examples. Consequently it is impossible to verify whether the bound is non-vacuous when the selected submatrix eigenvalues are only marginally positive or when the effective condition number of A is large.

    Authors: The complete derivation appears in the proof of the main theorem (Section 3). Because the result is a deterministic matrix inequality, no probabilistic error analysis is needed. The numerical examples in Section 4 already cover several indefinite cases, but we acknowledge that they do not explicitly test marginal positivity of the submatrix eigenvalues or large effective condition numbers of A. In the revision we will add a dedicated tightness subsection containing two new examples that address these regimes and confirm that the bound remains non-vacuous. revision: partial

Circularity Check

0 steps flagged

No significant circularity; bound uses independent matrix invariants

full rationale

The claimed eigenvalue lower bound for the Hadamard product is expressed in terms of the rank, effective condition number, and diagonal entries of one factor together with the smallest eigenvalues of selected principal submatrices of the other factor. These quantities are standard, independently measurable properties of the input matrices and are not defined in terms of the bound itself. The abstract and description contain no self-referential fitting, no renaming of known results as new derivations, and no load-bearing self-citations that reduce the central claim to a tautology. The selection of submatrices is presented as part of the statement of the result rather than as a fitted or circular choice. This is the normal case of a self-contained matrix inequality.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities beyond standard notions of rank, eigenvalues, and positive semidefiniteness.

pith-pipeline@v0.9.0 · 5377 in / 1118 out tokens · 40464 ms · 2026-05-10T01:43:35.639307+00:00 · methodology

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