﻿ 两个新的张量特征值包含区域 Two New Eigenvalue Inclusion Sets for Tensors

Pure Mathematics
Vol.06 No.05(2016), Article ID:18557,9 pages
10.12677/PM.2016.65055

Two New Eigenvalue Inclusion Sets for Tensors

Ruiyan Hu, Jing Zhao, Yaotang Li*

School of Mathematics and Statistics, Yunnan University, Kunming Yunnan

Received: Sep. 1st, 2016; accepted: Sep. 16th, 2016; published: Sep. 20th, 2016

ABSTRACT

The concept of tensors is a generalization of matrices to high order. And there are some important applications in many scientific fields, such as data analysis, signal and image processing and so on. Tensor eigenvalue theory is an important aspect of tensor research and application. In this paper, two new eigenvalue inclusion sets for tensors are given, and it is proved that the new eigenvalue inclusion sets are tighter than the classical Gersgorin inclusion set. In addition, as applications of the results, two sufficient conditions for the (semi-)positive definite property of the even order symmetric tensors are obtained.

Keywords:Tensor, Eigenvalue Inclusion Set, Symmetric Tensor, Positive Definite

1. 引言

，若，其中，则称为m阶n维的复(实)张量，记作。显然，向量是一阶张量，矩阵是二阶张量。此外，若存在复(实)和非零向量满足多元齐次方程：

，若的所有特征值都不为零，则称是非奇异的。称为Z-张量，如果它的所有非对角元非正，即可表示为，其中 ( [6] ，定义3)。此外，若，则称为M-张量；若，则称为非奇异的M-张量( [6] ，定义4)。

，定义次齐次元多项式如下：

2015年，Li等在文 [9] 中又给出张量特征值的如下包含区域。

2. 新的张量特征值包含区域

。 (1)

，则。由(1)式得：

，故

(2)

。(3)

。 (4)

。 (5)

，即或，因而。当时，由(5)式得

3. 数值例子

4. 偶数阶实对称张量正定性的判定

(6)

。 (7)

Figure 1. The comparison of ,

Figure 2.The comparison of

Figure 3.The comparison of

Two New Eigenvalue Inclusion Sets for Tensors[J]. 理论数学, 2016, 06(05): 402-410. http://dx.doi.org/10.12677/PM.2016.65055

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*通讯作者。