Sharp Spectral Bounds for Symmetric Positive Definite Tensors via Multiple Algebraic Invariants

Jul 16, 2026

Extends the trace–determinant framework for bounding H-eigenvalues of symmetric positive definite tensors. Replaces the AM–GM relaxation with the exact solution of a constrained Lagrangian optimization problem, producing sharp upper and lower bounds achieved on the admissible spectral variety. Adds higher-order power sums as spectral invariants and proves a structural theorem: any extremizer on a K-invariant feasible region has at most K distinct spectral values, reducing the search to finite low-dimensional polynomial systems. For the four-invariant case, provides complete theory including solution counts, a multistart Newton method, sharpness conditions, closed-form small-dimensional bounds, perturbation estimates, and refined Lyapunov region-of-attraction bounds. Numerical experiments up to d=100 show reduced median relative overestimation from 53% to 6%.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages24

Reading Time1 h 0 min

Introduction

H-eigenvalue problem and background

Limitations of prior trace–determinant/AM–GM approach

Contributions of this paper

Sharp two-invariant bound

Sharp three-invariant bound

General K-invariant hierarchy

Four-invariant case: (T, S, p3, D)

Closed-form bounds for small dimensions

Perturbation analysis

arXiv :2607 .08113v1 [math .OC] 9 Jul 2026 Sharp Spectral Bounds for Symmetric Positive Definite Tensors via Multiple Algebraic Invariants Hemant Sharma∗1, Snigdhashree Nayak†2, and Ankit Singh‡3 1 Department of Mathematics, Indian Institute of Information Technology, Design and Manufacturing Kancheepuram-600127, India 2 Department of Mathematics, Ravenshaw university, Odisha-753003, India 3 Department of Applied Sciences and Humanities, SVKM’s College of Engineering, Shirpur-425405, India Abstract We extend the trace–determinant framework of Nayak, Sharma, and Mishra [1] for bounding the H-eigenvalues of symmetric positive definite tensors. First, we replace the Arithmetic– Geometric Mean (AM–GM) relaxation underlying previous bounds by the exact solution of the associated constrained optimization problem, yielding sharp upper and lower bounds that are attained on the admissible spectral variety. Second, we incorporate higher-order power sums as additional spectral invariants and prove a structural theorem showing that any extremizer over a K-invariant feasibility region has at most K distinct spectral values. This reduces the problem to a finite collection of low-dimensional polynomial systems and yields a hierarchy of increasingly tight bounds. For the four-invariant case (T, S, p3 , D), we develop a complete theory including solution-count estimates, a multistart Newton algorithm, and sharpness conditions. We also derive closed-form bounds in small dimensions, establish perturbation estimates, and obtain refined Lyapunov region-of-attraction bounds. Numerical experiments for dimensions up to d = 100 show that the sharp three-invariant bound reduces the median relative overestimation gap from 53% to 6% while maintaining low computational cost. The framework is validated on tensors with real H-spectrum. Keywords: Eigenvalue bounds, symmetric tensors, power sums, Lagrangian extremization, Lyapunov stability, region of attraction. MSC2020: 15A18, 15A69, 15A42, 65F15, 93D05 . 1 Introduction The H-eigenvalue problem for a real mth order n-dimensional symmetric tensor A ∈ R [n,m] is the nonlinear system Axm−1 = λx [m−1], with x [m−1] := (xm1−1 , , xmn−1)T . Introduced independently by Lim [2] and Qi [3], this problem now plays a central role in higher-order data analysis [7], automatic control [5], diffusion tensor imaging [4], spectral hypergraph theory [12], and quantum entanglement detection [14] . In a recent contribution, Nayak, Sharma, and Mishra [1] introduced an algebraic framework for bounding the spectral radius and the smallest eigenvalue of a symmetric positive definite tensor ∗ [sharmahemant39@gmail.com](sharmahemant39@gmail.com)[ ](sharmahemant39@gmail.com)† [snigdhashreenayak91@gmail.com](snigdhashreenayak91@gmail.com)[ ](snigdhashreenayak91@gmail.com)‡Corresponding [author.as92393@gmail.com](author.as92393@gmail.com) by leveraging two intrinsic invariants, the trace tr(A) and the determinant det(A) (defined via the resultant of Axm−1 = 0) . Through repeated application of the AM–GM inequality, they derived a hierarchy of bounds (their Theorems 3.1–3.6) that strictly dominates the classical Gershgorin circle bounds, particularly when the tensor has negative off-diagonal entries or when the order m is large. Two limitations. Two structural limitations of that work motivate the present article. (L1) AM–GM is a relaxation. The AM–GM inequality replaces an exact constrained extremization by a tractable but loose surrogate. The actual optimization problem max 􀀈λ1 : P λi = T, Q λi = D, λi > 0 􀀉 admits a closed-form solution via Lagrange multipliers. The bound thus obtained is sharp on the spectral variety, while the AM–GM bound can be strictly slacker. (L2) Only two invariants are used. The trace and determinant are merely two coordinates of an infinite family of algebraic invariants of A, namely the power sums pk(A) = Pdi=1 λki , k = 1 , 2 ,... , each of which is an absolute invar

FAQs

What replaces the AM–GM relaxation in the new eigenvalue bounds?

The approach replaces AM–GM with the exact solution of the associated constrained optimization problem using Lagrangian extremization, producing bounds that are sharp on the admissible spectral variety.

How do additional power sums improve the bounds for tensor eigenvalues?

Higher-order power sums are included as extra algebraic invariants, tightening the feasibility region in the constrained optimization and leading to a hierarchy of increasingly accurate bounds.

What does the structural theorem say about an extremal spectrum?

Over any K-invariant feasible region, the extremizer spectrum contains at most K distinct spectral values, reducing the optimization to solving a finite collection of low-dimensional polynomial systems.

Jul 16, 2026

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