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Symmetric positive definite tensors admit eigenvalue bounds from their trace and resultant via AM-GM that stay tight when Gershgorin fails.

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2026-06-30 20:27 UTC pith:4KUDQVRV

load-bearing objection The paper gives usable coordinate-free eigenvalue bounds for SPD tensors via AM-GM on trace and resultant, with better behavior than Gershgorin in the highlighted cases.

arxiv 2605.14768 v1 pith:4KUDQVRV submitted 2026-05-14 math.NA cs.NA

Eigenbounds of symmetric positive definite tensors

Snigdhashree Nayak , Hemant Sharma , Nachiketa Mishra This is my paper
classification math.NA cs.NA
keywords eigenvalue boundssymmetric tensorspositive definite tensorsAM-GM inequalityGershgorin circle theoremspectral radiusresultantLyapunov stability
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes a coordinate-free method for bounding the largest and smallest eigenvalues of symmetric positive definite tensors by applying the AM-GM inequality directly to the tensor's trace and resultant. This produces a hierarchy of upper and lower bounds on the spectral radius and minimal eigenvalue without needing the entries in any particular basis. A sympathetic reader would care because the same invariants also certify positive definiteness of Lyapunov functions, supplying a practical test for stability of nonlinear systems where classical entry-wise bounds become uninformative.

Core claim

By treating the trace and the resultant as intrinsic invariants of a symmetric positive definite tensor, the AM-GM inequality generates a nested family of bounds on the spectral radius and the smallest eigenvalue; these bounds remain valid and informative precisely when coordinate-dependent estimates such as the Gershgorin circle theorem lose tightness because of negative off-diagonal entries or combinatorial explosion in higher-order cases. The same bounds are shown to certify positive definiteness of candidate Lyapunov functions for autonomous nonlinear systems.

What carries the argument

Hierarchy of AM-GM inequalities applied to the trace and resultant invariants of the tensor.

Load-bearing premise

The tensors are symmetric and positive definite, so their trace and resultant serve as reliable invariants to which AM-GM can be applied without coordinate information.

What would settle it

For a symmetric positive definite tensor with at least one negative off-diagonal entry, compute its exact eigenvalues and check whether the AM-GM bounds are both valid and narrower than the Gershgorin intervals.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The bounds supply an algebraic certificate of positive definiteness without solving the characteristic equation.
  • The hierarchy produces successively tighter estimates by incorporating higher powers of the invariants.
  • The method remains informative for tensors of order greater than three where entry-wise counting becomes prohibitive.
  • Direct application to Lyapunov functions yields stability conclusions for nonlinear systems from trace and resultant data alone.

Where Pith is reading between the lines

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

  • The same invariant approach might be tested on tensors that are only weakly positive definite to quantify how close the bounds come to the true spectral edge.
  • Replacing AM-GM with other symmetric-mean inequalities could generate alternative bound families for the same invariants.
  • The method's coordinate independence suggests it could serve as a preconditioner or quick check inside numerical eigensolvers for large tensors.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The manuscript introduces an algebraic framework for bounding the eigenvalues of symmetric positive definite tensors by applying the AM-GM inequality to the trace and resultant (determinant) as coordinate-free invariants. It derives a hierarchy of progressively tighter upper and lower bounds on the spectral radius and smallest eigenvalue, presents a comparative analysis against the Gershgorin circle theorem (claiming superior robustness for negative off-diagonal entries and higher-order tensors), and applies the bounds to certify positive definiteness of Lyapunov functions in nonlinear autonomous system stability analysis.

Significance. If the derivations hold, the invariant-based approach offers a potentially useful coordinate-free alternative for eigenvalue estimation in tensor problems where classical methods suffer from combinatorial growth or cancellations. The explicit robustness claims for cases where Gershgorin fails and the Lyapunov application provide practical context. The construction is internally consistent with the definition of SPD tensors and leverages standard inequalities without introducing free parameters or self-referential quantities.

minor comments (3)
  1. [§3] §3 (hierarchy derivation): the transition from the basic AM-GM on trace and resultant to the full hierarchy of bounds should include an explicit inductive step or recurrence relation to make the progressive tightening transparent.
  2. [Table 2] Table 2 (comparative examples): the reported bound ratios for order-4 tensors with negative entries would be strengthened by including the exact tensor entries or a reproducible construction method rather than summary statistics alone.
  3. [§5] §5 (Lyapunov application): the certification step assumes the tensor is exactly SPD; a brief remark on how the bounds behave under numerical perturbation of the invariants would clarify robustness for practical use.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its potential utility as a coordinate-free approach, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation applies the standard AM-GM inequality directly to the trace (sum of eigenvalues) and resultant (product of eigenvalues) of symmetric positive definite tensors, which are intrinsic invariants by definition of the tensor class. No equations reduce a claimed prediction or bound back to a fitted parameter or self-referential definition; the comparison to Gershgorin is an external benchmark rather than a load-bearing premise. The approach is self-contained against the mathematical properties of SPD tensors and does not rely on self-citations or ansatzes imported from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard AM-GM inequality and the algebraic properties of symmetric positive definite tensors; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)
  • standard math AM-GM inequality applies to positive real numbers derived from tensor invariants
    Invoked to generate the hierarchy of bounds on spectral radius and smallest eigenvalue.

pith-pipeline@v0.9.1-grok · 5672 in / 1292 out tokens · 28900 ms · 2026-06-30T20:27:19.826845+00:00 · methodology

0 comments
read the original abstract

This article introduces an algebraic framework for establishing eigenvalue bounds for symmetric positive definite tensors by leveraging intrinsic invariants, specifically the trace and determinant (resultant). We derive a hierarchy of inequalities via the Arithmetic Mean-Geometric Mean (AM-GM) inequality that yields progressively tighter upper and lower bounds for the tensor spectral radius and smallest eigenvalue. A comprehensive comparative analysis demonstrates that our invariant-based approach significantly outperforms classical coordinate-dependent methods such as the Gershgorin circle theorem. We explicitly show that our bounds remain robust and informative in scenarios where Gershgorin bounds fail, particularly for tensors with negative off-diagonal entries, where algebraic cancellations occur, and higher-order tensors, where combinatorial growth leads to loose estimates. Furthermore, we validate the practical utility of these bounds by applying them to certify the positive definiteness of Lyapunov functions in the stability analysis of nonlinear autonomous systems.

Figures

Figures reproduced from arXiv: 2605.14768 by Hemant Sharma, Nachiketa Mishra, Snigdhashree Nayak.

Figure 1
Figure 1. Figure 1: Hierarchy of Upper Bounds for Example 5.1. The plot visually demonstrates the convergence [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Eigenvalue Distribution vs Bounds for Example 5.1. The plot displays the actual eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Spectral Interval Comparison for Example 5.1. The bars represent the guaranteed range [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bound Comparison for 6th-Order Tensor. The Gershgorin bound explodes to 70.00 due to [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Eigenvalue Distribution vs Bounds for Example 5.2 ( [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spectral Interval Comparison for Example 5.2. The Gershgorin interval (Red) extends deeply [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗

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Reference graph

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