Positive Definiteness of $4$th Order $3$-Dimensional Symmetric Tensors with entries $-1$, $0$, $1$

TL;DR

This work tackles the problem of determining positive definiteness for 4th-order, 3-dimensional symmetric tensors with entries drawn from the set $\{-1,0,1\}$. By representing the tensor action as a quartic homogeneous polynomial $\mathcal{T}x^4$ in three variables and employing sum-of-squares decompositions together with a systematic partition of entries (notably groups like $t_{1112}t_{1222}$ and $t_{1122}$), the authors derive a suite of necessary and sufficient criteria for PSD and PD. The results are organized into a sequence of theorems (e.g., Theorems on PSD/PD conditions for specific diagonal patterns and off-diagonal blocks) and culminate in corollaries that unify the 3D case with known 2D and prior 3D tensor theory. The findings reveal rich, case-dependent structures that distinguish higher-order tensors from matrices and provide explicit, verifiable certificates for positivity, with potential implications for higher-order optimization and tensor copositivity analysis.

Abstract

It is well-known that a symmetric matrix with its entries $\pm1$ is not positive definite. But this is not ture for symmetric tensors (hyper-matrix). In this paper, we mainly dicuss the positive (semi-)definiteness criterion of a class of $4$th order $3$-dimensional symmetric tensors with entries $t_{ijkl}\in\{-1,0,1\}$. Through theoretical derivations and detailed classification discussions, the criterion for determining the positive (semi-)definiteness of such a class of tensors are provided based on the relationships and number values of its entries. Which establishes some unique properties of higher symmetric tensors that distinct from ones of matrces

Theorems & Definitions (37)

• Definition 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Theorem 3.1
• proof