Eigenbounds of symmetric positive definite tensors

TL;DR

This paper develops an algebraic method using tensor invariants to derive tight eigenvalue bounds for symmetric positive definite tensors, outperforming classical methods in robustness and applicability.

Contribution

It introduces a hierarchy of invariant-based inequalities for eigenvalue bounds, improving over traditional coordinate-dependent techniques like Gershgorin.

Findings

Invariant-based bounds outperform Gershgorin bounds for tensors with negative off-diagonal entries.

Bounds remain robust in higher-order tensors and scenarios with algebraic cancellations.

Application to Lyapunov functions aids stability analysis of nonlinear systems.

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.