计算几类3阶对称张量特征值的直接方法

Direct Methods for Calculating Several Clas-ses of Eigenvalues of 3th Order Symmetric Tensors

协正张量是一种重要的结构张量,在许多领域都有着广泛的应用,成为近年来新兴的研究课题。已有研究表明,对称张量是严格协正的当且仅当其所有Pareto-H特征值是正的,而Pareto-H特征值与H++-特征值又具有一定的联系。另外,对张量特征值计算的研究是张量理论研究的一个重要部分。因此,求出对称张量特征值的具体表达式是很有必要的。本文主要介绍了计算几类3阶对称张量特征值的直接方法。首先,给出了计算3阶2维对称张量的H+-特征值的直接方法,分别建立了3阶2维对称张量的H+-特征值、H++-特征值以及Pareto H-特征值的具体表达式。然后利用张量的Pareto H-特征值与协正性之间的关系,给出了判定3阶2维对称张量协正性的充分条件。

The copositive tensors is an important structural tensors, which has been widely used in many fields and has become an emerging research topic in recent years. Previous studies have shown that a symmetric tensor is strictly copositive if and only if each of its Pareto-H eigenvalue is positive, and the Pareto-H eigenvalue have a certain relationship with the H++-eigenvalue. In addition, the study of computing tensor eigenvalues is an important part of tensors theory. Therefore, it is necessary to find the precise expressions of the eigenvalues of the symmetric tensors. In this paper, we mainly introduce some direct methods for calculating several classes of eigenvalues of 3th order symmetric tensors. First of all, we in this paper propose a direct method for calculating all H+-eigenvalue of 3th order 2 dimensional symmetric tensors, and the expressions of the H+-eigenvalue, the H++-eigenvalue and the Pareto H-eigenvalue of such tensors are established. Then, using the relationship between the Pareto H-eigenvalue and copositivity of tensors, we obtain analytically sufficient conditions for determining copositivity of 3th order 2 dimensional symmetric tensors.