贵州民族大学数据科学与信息工程学院，贵州 贵阳

收稿日期：2022年7月15日；录用日期：2022年8月9日；发布日期：2022年8月17日

摘要

H-张量在科学与工程实践等领域中有着重要的应用，但在实际中要判定H-张量是比较困难的。本文通过构造不同的正对角阵，结合不等式的放缩技巧，给出了H-张量比较实用的新判别条件。作为应用，给出了判定偶次齐次多项式正定性的新方法，并给出相应的数值算例，表明了新结论的有效性。

关键词

H-张量，齐次多项式，正定性，不可约，非零元素链

New Criterions for H-Tensors and Its Applications

Pengcheng Zhao

College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang Guizhou

Received: Jul. 15^th, 2022; accepted: Aug. 9^th, 2022; published: Aug. 17^th, 2022

ABSTRACT

H-tensors have important applications in science and engineering, but it is difficult to determine whether a given tensor is an H-tensor or not in practice. In this paper, by constructing different positive diagonal matrices and combining the technique of inequality reduction, new practical conditions for H-tensors are given. As applications, new methods for determining the positive definiteness of even homogeneous polynomials are presented, and the validity of new results is verified by some numerical examples.

Keywords:H-Tensors, Homogeneous Multivariate Forms, Positive Definiteness, Irreducible, Nonzero Elements Chain

Copyright © 2022 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

张量是高阶广义矩阵，广泛应用于信号和图像处理、高阶统计学、自动控制、医学成像、超图理论、弹性材料科学和工程研究与数据分析等领域中。近年来，许多专家学者对一般张量 [1] - [6] 或特殊结构张量的理论、性质及应用进行了广泛探讨 [7] - [18]。本文继续讨论H-张量的判定问题，得到了只与张量元素有关的新判别不等式，拓广了文献 [11] [14] [15] [16] 的结果。同时，获得了偶数阶实对称张量，即偶数阶齐次多项式正定性的新判定条件。最后，利用数值算例说明了新条件的有效性。

2. 预备知识

记 $C\left(R\right)$ 为复(实)数集， $N=\left[n\right]=\left\{1,2,\cdots ,n\right\}$。一个m阶n维张量 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 由 $n^m$ 个元素构成，其中 $a_{i_1}{{}_{i_2\cdots }}_{i_m}\in C$，$i_j\in N$，$j\in \left[m\right]$ [3] [4]。若 $a_{i_1{i}_2\cdots i_m}=a_{\pi \left(i_1{i}_2\cdots i_m\right)}$，$\forall \pi \in {\Pi }_m$，则称 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 为对称张量 [5]，其中 ${\Pi }_m$ 为m个指标的置换群。称张量 $I=\left({\delta }_{i_1{i}_2\cdots i_m}\right)$ 为单位张量 [5]，若

${\delta }_{i_1{i}_2\cdots i_m}\text{=}\{\begin{array}{l}1,i_1=i_2=\cdots =i_m,\\ 0,其它.\end{array}$

若

$f\left(x\right)={\displaystyle \underset{i_1,\cdots ,i_m\in \left[n\right]}{\sum }{a}_{i_1{i}_2\cdots i_m}{x}_{i_1}\cdots x_{i_m}}>0$，$x={\left(x_1,x_2,\cdots ,x_n\right)}^T\in R^n$，$x\ne 0$，

则称m阶n次齐次多项式 $f\left(x\right)$ 是正定的 [3]。 $f\left(x\right)$ 也可以表示为m阶n维对称张量 $A$ 与 $x^m$ 的乘积，如下

$f\left(x\right)=A{x}^m={\displaystyle \underset{i_1,\cdots ,i_m\in \left[n\right]}{\sum }{a}_{i_1{i}_2\cdots i_m}{x}_{i_1}\cdots x_{i_m}}$.

若 $f\left(x\right)$ 是正定的，则对称张量 $A$ 也是正定的 [3]。

定义1 [8] 设 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 是m阶n维张量，若存在正向量 $x={\left(x_1,x_2,\cdots ,x_n\right)}^T\in R^n$，满足

$\left|a_{ii\cdots i}\right|x_i^{m-1}>{\displaystyle \underset{\begin{array}{c}{i}_2,i_3,\cdots ,i_m\in \left[n\right]\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}\cdots x_{i_m}}$，$\forall i\in N$，

则称 $A$ 为H-张量。

定义2 [5] 设 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 是m阶n维张量，若存在一个非空子集 $I\subset N$，满足

$a_{i_1{i}_2\cdots i_m}=0$，$\forall i_1\in I$，$\forall i_2,\cdots ,i_m\notin I$，

则称 $A$ 是可约张量。否则，称 $A$ 是不可约张量。

定义3 [9] 设 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 是m阶n维张量，若存在指标 $k_1,k_2,\cdots ,k_r$，满足

${\displaystyle \underset{\begin{array}{c}{i}_2,i_3,\cdots ,i_m\in \left[n\right]\\ {\delta }_{k_s{i}_2\cdots i_m}=0\\ k_{s+1}\in \left\{i_2,i_3,\cdots ,i_m\right\}\end{array}}{\sum }\left|a_{k_s{i}_2\cdots i_m}\right|}\ne 0$，$s=0,1,\cdots ,r$，$\forall i,j\in N\left(i\ne j\right)$，

其中 $k_0=i$，$k_{r+1}=j$，则称张量 $A$ 中有一条从指标i到指标j的非零元素链。

3. 主要结果

为讨论方便，给出如下记号：设 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 是m阶n维张量，令

$S^{m-1}=\left\{i_2{i}_3\cdots i_m:i_j\in S,j=2,3,\cdots ,m\right\}$，

$N^{m-1}\backslash S^{m-1}=\left\{i_2{i}_3\cdots i_m:i_2{i}_3\cdots i_m\in N^{m-1}且i_2{i}_3\cdots i_m\notin S^{m-1}\right\}$，

$r_i\left(A\right)={\displaystyle \underset{\begin{array}{c}{i}_2,\cdots ,i_m\in \left[n\right]\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}={\displaystyle \underset{i_2,\cdots ,i_m\in \left[n\right]}{\sum }\left|a_{i{i}_2\cdots i_m}\right|-\left|a_{ii\cdots i}\right|}$，

$N_1=\left\{i\in N:0<\left|a_{ii\cdots i}\right|=r_i\left(A\right)\right\}$，$N_2=\left\{i\in N:0<\left|a_{ii\cdots i}\right|<r_i\left(A\right)\right\}$，

$N_3=\left\{i\in N:\left|a_{ii\cdots i}\right|>r_i\left(A\right)\right\}$，$N_0^{m-1}=N^{m-1}/\left(N_2^{m-1}\cup N_3^{m-1}\right)$，

${\beta }_i=\frac{r_i\left(A\right)-\left|a_{ii\cdots i}\right|}{r_i\left(A\right)}$，$M=\underset{\begin{array}{c}i\in N_2\\ j\in N_3\end{array}}{\max }\left\{{\beta }_i,\frac{r_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\right\}$，

$K=\underset{i\in N_3}{\max }\frac{M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}+{\displaystyle \underset{i_2\cdots i_m\in N_2^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}}{r_i\left(A\right)-{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_3^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}}$，

$R_i\left(A\right)=M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}+{\displaystyle \underset{i_2\cdots i_m\in N_2^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}+K{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_3^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}\left(i\in N_3\right)$.

引理1 [6] 若 $A$ 是严格对角占优的张量，则 $A$ 是H-张量。

引理2 [10] 设 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 是m阶n维张量。若存在正对角阵 $X$，满足 $A{X}^{m-1}$ 是H-张量，则 $A$ 是H-张量。

引理3 [6] 设 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 是m阶n维张量且不可约。若

$\left|a_{ii\cdots i}\right|\ge r_i\left(A\right)$，$\forall i\in N$，

且上式中至少有一个严格不等式成立，则 $A$ 是H-张量。

引理4 [9] 设 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 是m阶n维张量。若

1) $\left|a_{ii\cdots i}\right|\ge r_i\left(A\right)$，$\forall i\in N$ ；

2) $J\left(A\right)=\left\{i\in N:\left|a_{ii\cdots i}\right|>r_i\left(A\right)\right\}\ne \varnothing$ ；

3) $\forall i\notin J\left(A\right)$，从指标i到指标j有一条非零元素链，满足 $j\in J\left(A\right)$ ；

则 $A$ 是H-张量。

定理1 设 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 是m阶n维张量。若 $A$ 满足

$\begin{array}{c}\left|a_{ii\cdots i}\right|{\beta }_i>M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}+{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\right\}\left|a_{i{i}_2\cdots i_m}\right|}\end{array}$，$\forall i\in N_2$， (1)

且 $\left|a_{ii\cdots i}\right|\ne {\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_0^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}$ ( $\forall i\in N_1$ )，则 $A$ 是H-张量。

证明：由K的定义知

$K\left|a_{ii\cdots i}\right|\ge M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}+{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}}\left|a_{i{i}_2\cdots i_m}\right|+{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}=R_i\left(A\right)$.

因此

$K\ge \frac{R_i\left(A\right)}{\left|a_{ii\cdots i}\right|}$，$\forall i\in N_3$. (2)

由式(1)得，

$\left|a_{ii\cdots i}\right|{\beta }_i-M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}-{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}-{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\right\}\left|a_{i{i}_2\cdots i_m}\right|}>0$. (3)

而 $R_i\left(A\right)<r_i\left(A\right)$ 且 $M\ge \frac{r_i\left(A\right)}{\left|a_{ii\cdots i}\right|}\left(\forall i\in N_3\right)$，所以

$M>\frac{R_i\left(A\right)}{\left|a_{ii\cdots i}\right|}$，$\forall i\in N_3$. (4)

由(3)式和(4)式得，一定存在足够小的正数 $\epsilon$，使得

$M>\frac{R_i\left(A\right)}{\left|a_{ii\cdots i}\right|}+\epsilon$，$\forall i\in N_3$， (5)

$\begin{array}{l}\left|a_{ii\cdots i}\right|{\beta }_i-M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}-{\displaystyle \underset{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2\cdots i_m\in N_2^{m-1}}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ -{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\right\}\left|a_{i{i}_2\cdots i_m}\right|}>\epsilon {\displaystyle \underset{i_2\cdots i_m\in N_3^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}\end{array}$，$\forall i\in N_2$. (6)

构造正对角阵 $X=diag\left(x_1,x_2,\cdots ,x_n\right)$，记，其中

$x_i=M$，$i\in N_1$ ； $x_i={\beta }_i^{\frac{1}{m-1}}$，$i\in N_2$ ； $x_i={\left(\frac{R_i\left(A\right)}{\left|a_{ii\cdots i}\right|}+\epsilon \right)}^{\frac{1}{m-1}}$，$i\in N_3$.

而 $M\ge \frac{r_i\left(A\right)-\left|a_{ii\cdots i}\right|}{r_i\left(A\right)}\left(i\in N_2\right)$ 且 $M>\frac{r_i\left(A\right)}{\left|a_{ii\cdots i}\right|}+\epsilon \left(i\in N_3\right)$，故对 $\forall i\in N_1$，

$\begin{array}{c}{r}_i\left(B\right)={\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_0^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}}\cdots x_{i_m}+{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_2^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\beta }_{i_2}^{\frac{1}{m-1}}\cdots }{\beta }_{i_m}^{\frac{1}{m-1}}\\ +{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_3^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\left(\frac{R_{i_2}\left(A\right)}{\left|a_{i_2{i}_2\cdots i_2}\right|}+\epsilon \right)}^{\frac{1}{m-1}}\cdots }{\left(\frac{R_{i_m}\left(A\right)}{a_{i_m{i}_m\cdots i_m}}+\epsilon \right)}^{\frac{1}{m-1}}\\ \le M{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_0^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|+}{\displaystyle \underset{i_2\cdots i_m\in N_2^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}+\epsilon \right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ \le M{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_0^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}+M{\displaystyle \underset{i_2\cdots i_m\in N_2^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}+M{\displaystyle \underset{i_2\cdots i_m\in N_3^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}\\ =M{r}_j\left(A\right)=M\left|a_{ii\cdots i}\right|=\left|b_{ii\cdots i}\right|.\end{array}$

根据(6)式，对 $\forall i\in N_2$，

$\begin{array}{c}{r}_i\left(B\right)={\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}}\cdots x_{i_m}+{\displaystyle \underset{\begin{array}{c}{i}_2{i}_3\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\beta }_{i_2}^{\frac{1}{m-1}}\cdots }{\beta }_{i_m}^{\frac{1}{m-1}}\\ +{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_3^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\left(\frac{R_{i_2}\left(A\right)}{\left|a_{i_2{i}_2\cdots i_2}\right|}+\epsilon \right)}^{\frac{1}{m-1}}\cdots }{\left(\frac{R_{i_m}\left(A\right)}{a_{i_m{i}_m\cdots i_m}}+\epsilon \right)}^{\frac{1}{m-1}}\\ \le M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|+}{\displaystyle \underset{\begin{array}{c}{i}_2{i}_3\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}+\epsilon {\displaystyle \underset{i_2\cdots i_m\in N_3^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}\end{array}$

$\begin{array}{l}<M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|+}{\displaystyle \underset{\begin{array}{c}{i}_2{i}_3\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}+\left|a_{ii\cdots i}\right|{\beta }_i-M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}\\ -{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}-{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ =\left|a_{ii\cdots i}\right|{\beta }_i=\left|b_{ii\cdots i}\right|.\end{array}$

对 $\forall i\in N_3$，由(2)式得

$\begin{array}{c}{r}_i\left(B\right)={\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}}\cdots x_{i_m}+{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_2^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\beta }_{i_2}^{\frac{1}{m-1}}\cdots {\beta }_{i_m}^{\frac{1}{m-1}}}\\ +{\displaystyle \underset{\begin{array}{c}{i}_2{i}_3\cdots i_m\in N_3^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\left(\frac{R_{i_2}\left(A\right)}{\left|a_{i_2{i}_2\cdots i_2}\right|}+\epsilon \right)}^{\frac{1}{m-1}}\cdots }{\left(\frac{R_{i_m}\left(A\right)}{a_{i_m{i}_m\cdots i_m}}+\epsilon \right)}^{\frac{1}{m-1}}\\ \le M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|+}{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_2^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_{\text{3}}^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}+\epsilon {\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_3^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}\end{array}$

$\begin{array}{l}\le M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|+}{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_2^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_{\text{3}}^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }K\left|a_{i{i}_2\cdots i_m}\right|}+\epsilon {\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_3^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}\\ =R_i\left(A\right)+\epsilon {\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_3^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}<R_i\left(A\right)+\epsilon {\displaystyle \underset{i_2\cdots i_m\in N_3^{m-1}}{\sum }\left|a_{ii\cdots i}\right|}\\ =\left|a_{ii\cdots i}\right|\left(\frac{R_i\left(A\right)}{\left|a_{ii\cdots i}\right|}+\epsilon \right)=\left|b_{ii\cdots i}\right|.\end{array}$

综上可得 $\left|b_{ii\cdots i}\right|>r_i\left(B\right)\left(\forall i\in N\right)$。由引理1知 $B$ 是H-张量，故由引理2知 $A$ 是H-张量。

定理2 设 $A=\left(a_{i_1{i}_2\cdots i_m}\right)$ 是m阶n维张量且不可约。若 $A$ 满足

$\begin{array}{c}\left|a_{ii\cdots i}\right|{\beta }_i\ge M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}+{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\right\}\left|a_{i{i}_2\cdots i_m}\right|}\end{array}$，$\forall i\in N_2$， (7)

且(7)式中至少有一个严格不等式成立，则 $A$ 是H-张量。

证明：构造正对角阵 $X=diag\left(x_1,x_2,\cdots ,x_n\right)$，记 $B=A{X}^{m-1}=\left(b_{i_1{i}_2\cdots i_m}\right)$，其中

$x_i=M$，$i\in N_1$ ； $x_i={\beta }_i^{\frac{1}{m-1}}$，$i\in N_2$ ； $x_i={\left(\frac{R_i\left(A\right)}{\left|a_{ii\cdots i}\right|}\right)}^{\frac{1}{m-1}}$，$i\in N_3$.

由M的定义得，对 $\forall i\in N_1$，

$\begin{array}{c}{r}_i\left(B\right)={\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_0^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}}\cdots x_{i_m}+{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_2^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\beta }_{i_2}^{\frac{1}{m-1}}\cdots {\beta }_{i_m}^{\frac{1}{m-1}}}\\ +{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_3^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\left(\frac{R_{i_2}\left(A\right)}{\left|a_{i_2{i}_2\cdots i_2}\right|}\right)}^{\frac{1}{m-1}}\cdots }{\left(\frac{R_{i_m}\left(A\right)}{a_{i_m{i}_m\cdots i_m}}\right)}^{\frac{1}{m-1}}\\ \le M{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_0^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|+}{\displaystyle \underset{i_2\cdots i_m\in N_2^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ \le M{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_0^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|+}M{\displaystyle \underset{i_2\cdots i_m\in N_2^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}+M{\displaystyle \underset{i_2\cdots i_m\in N_3^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|}\\ =M{r}_j\left(A\right)=M\left|a_{ii\cdots i}\right|=\left|b_{ii\cdots i}\right|.\end{array}$

根据(7)式知，对 $\forall i\in N_2$，

$\begin{array}{c}{r}_i\left(B\right)={\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}}\cdots x_{i_m}+{\displaystyle \underset{\begin{array}{c}{i}_2{i}_3\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\beta }_{i_2}{}^{\frac{1}{m-1}}\cdots }{\beta }_{i_m}{}^{\frac{1}{m-1}}\\ +{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_3^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\left(\frac{R_{i_2}\left(A\right)}{\left|a_{i_2{i}_2\cdots i_2}\right|}\right)}^{\frac{1}{m-1}}\cdots }{\left(\frac{R_{i_m}\left(A\right)}{a_{i_m{i}_m\cdots i_m}}\right)}^{\frac{1}{m-1}}\\ \le M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|+}{\displaystyle \underset{\begin{array}{c}{i}_2{i}_3\cdots i_m\in N_2^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{i_2\cdots i_m\in N_{\text{3}}^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}\\ \le \left|a_{ii\cdots i}\right|{\beta }_i=\left|b_{ii\cdots i}\right|.\end{array}$

又对 $\forall i\in N_3$，由K的定义知

$\begin{array}{c}{r}_i\left(B\right)={\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}}\cdots x_{i_m}+{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_2^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\beta }_{i_2}^{\frac{1}{m-1}}\cdots {\beta }_{i_m}^{\frac{1}{m-1}}}\\ +{\displaystyle \underset{\begin{array}{c}{i}_2{i}_3\cdots i_m\in N_3^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|{\left(\frac{R_{i_2}\left(A\right)}{\left|a_{i_2{i}_2\cdots i_2}\right|}\right)}^{\frac{1}{m-1}}\cdots }{\left(\frac{R_{i_m}\left(A\right)}{a_{i_m{i}_m\cdots i_m}}\right)}^{\frac{1}{m-1}}\\ \le M{\displaystyle \underset{i_2\cdots i_m\in N_0^{m-1}}{\sum }\left|a_{i{i}_2\cdots i_m}\right|+}{\displaystyle \underset{i_2{i}_3\cdots i_m\in N_2^{m-1}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\left\{{\beta }_j\right\}\left|a_{i{i}_2\cdots i_m}\right|}\\ +{\displaystyle \underset{\begin{array}{c}{i}_2\cdots i_m\in N_{\text{3}}^{m-1}\\ {\delta }_{i{i}_2\cdots i_m}=0\end{array}}{\sum }\underset{j\in \left\{i_2,i_3,\cdots ,i_m\right\}}{\max }\frac{R_j\left(A\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}\end{array}$