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

Abstract

It is well-known that a symmetric matrix with its entries $\pm 1$ 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

54 Pages