A general product of tensors with applications
A general product of tensors with applications
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张量的一般乘积及其应用
DOI：
10.1016/j.laa.2013.07.010
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发表时间：
2013-10-15
影响因子：
1.1
通讯作者：
Shao, Jia-Yu
中科院分区：
文献类型：
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作者：
Shao, Jia-Yu
本文研究了两个n维张量A和B的一般乘积，其中m >= 2，k >= 1.这种乘积满足结合律，是通常矩阵乘积的推广。利用这种积，张量的许多概念和已知结果可以被简单地表示和/或证明，并给出它的一些应用。利用这种张量积的结合律和n元齐次方程组结式的一些性质，定义了张量的相似性和同余性（它们也是矩阵对应关系的推广），证明了相似张量具有相同的特征多项式，从而具有相同的谱.研究了两种特殊的相似性：置换相似性和对角相似性，以及它们在超图谱和非负不可约张量谱研究中的应用。定义了张量的直积（在矩阵情况下也称为Kronecker积），并给出了它在研究超图的两类积的谱中的应用。我们也给出了这个一般乘积在非负张量研究中的应用，包括本原张量的一个特征，本原度的上界和一些非负不可约张量的循环指数. (C)2013 Elsevier Inc. All rights reserved.
We study a general product of two n-dimensional tensors A and B with orders m >= 2 and k >= 1. This product satisfies the associative law, and is a generalization of the usual matrix product. Using this product, many concepts and known results of tensors can be simply expressed and/or proved, and a number of applications of it will be given. Using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables, we define the similarity and congruence of tensors (which are also the generalizations of the corresponding relations for matrices), and prove that similar tensors have the same characteristic polynomials, thus the same spectra. We study two special kinds of similarity: permutational similarity and diagonal similarity, and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors. We also define the direct product of tensors (in matrix case it is also called the Kronecker product), and give its applications in the study of the spectra of two kinds of the products of hypergraphs. We also give applications of this general product in the study of nonnegative tensors, including a characterization of primitive tensors, the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors. (C) 2013 Elsevier Inc. All rights reserved.