New Criterions-Based $\mathcal{H}$-Tensors for Testing the Positive Definiteness of Multivariate Homogeneous Forms

Abstract

Positive definite homogeneous multivariate forms play an important role in polynomial problems and medical imaging, and the definiteness of forms can be tested using structured tensors. In this paper, we state the equivalence between the positive definite multivariate forms and the corresponding tensors, and explain the connection between the positive definite tensors with $\mathcal{H}$-tensors. Then, based on the notion of diagonally dominant tensors, some criteria for $\mathcal{H}$-tensors are presented. Meanwhile, with these links, we provide an iterative algorithm to test the positive definiteness of multivariate homogeneous forms and prove its validity theoretically. The advantages of the obtained results are illustrated by some numerical examples.

1. Introduction

Let $C\left(R\right)$ be the complex (real) field and $N=\{1,2,\cdots ,n\}$. $\mathcal{A}=\left(a_{i_1{i}_2\cdots i_m}\right)$ is called an mth-order n-dimensional complex (real) tensor [1,2], if

$$a_{i_1{i}_2\cdots i_m}\in C\left(R\right),$$

where $i_t=1,\dots ,n$ for $t=1,\dots ,m$. It is denoted by $\mathcal{A}\in C^{[m,n]}\left(R^{[m,n]}\right)$. Tensor $\mathcal{A}=\left(a_{i_1{i}_2\cdots i_m}\right)\in C^{[m,n]}$ is called symmetric [3], if

$$a_{i_1{i}_2\cdots i_m}=a_{\pi (i_1{i}_2\cdots i_m)},\forall \pi \in {\Pi }_m,$$

where ${\Pi }_m$ is the permutation group of m indices. Tensor $\mathcal{I}=\left({\delta }_{i_1{i}_2\cdots i_m}\right)\in C^{[m,n]}$ is called the unit tensor [4], if

$${\delta }_{i_1{i}_2\cdots i_m}=\left\{\begin{array}{cc}1,& \mathrm{if}{i}_1=\cdots =i_m,\\ 0,& \mathrm{otherwise}.\end{array}\right.$$

Tensor $\mathcal{A}$ is called a diagonally dominant tensor, if

$$|a_{ii\cdots i}|\ge \sum _{\genfrac{}{}{0pt}{}{i_2,\dots ,i_m\in N}{{\delta }_{i{i}_2\cdots i_m}=0}}|a_{i{i}_2\cdots i_m}|=R_i\left(\mathcal{A}\right),\forall i\in N.$$

(1)

$\mathcal{A}$ is strictly diagonally dominant if all strict inequalities in $\left(1\right)$ hold. If there exists a number $\lambda \in C$ and a non-zero vector $x={(x_1,x_2,\cdots ,x_n)}^T\in C^n$, such that

$$\mathcal{A}{x}^{m-1}=\lambda x^{[m-1]},$$

then $\lambda$ is called an eigenvalue of $\mathcal{A}$ and x is called the eigenvector of $\mathcal{A}$ associated with $\lambda$ [5,6,7], where $\mathcal{A}{x}^{m-1}$ and $\lambda x^{[m-1]}$ are vectors, whose ith components are

$${\left(\mathcal{A}{x}^{m-1}\right)}_i=\sum _{i_2,\dots ,i_m\in N}{a}_{i{i}_2\cdots i_m}{x}_{i_2}\cdots x_{i_m},{\left(x^{[m-1]}\right)}_i=x_i^{m-1}.$$

When $\lambda \in R$ and $x\in R^n$, we call $\lambda$ an H-eigenvalue of $\mathcal{A}$ and x an H-eigenvector of $\mathcal{A}$ associated with $\lambda$ [1].

An mth degree homogeneous polynomial of n variables $f\left(x\right)$ can be defined as

$$\begin{array}{c}\hfill f\left(x\right)\equiv \mathcal{A}{x}^m=\sum _{i_1,i_2,\dots ,i_m\in N}{a}_{i_1{i}_2\cdots i_m}{x}_{i_1}{x}_{i_2}\cdots x_{i_m},\end{array}$$

(2)

where $\mathcal{A}=\left(a_{i_1{i}_2\cdots i_m}\right)\in C^{[m,n]}$ is symmetric and $x=(x_1,x_2,\dots ,x_n)\in R^n$ [8]. When m is even, we call that $f\left(x\right)$ is positive definite if

$$f\left(x\right)>0,\forall x\in R^n,x\ne 0.$$

The tensor $\mathcal{A}$ is called positive definite if $f\left(x\right)$ is positive definite [8].

The positive definiteness of multivariate polynomial $f\left(x\right)$ plays an important role in automatical control and polynomial problems [9,10]. However, when $n\ge 4$ and $m\ge 4$, it is difficult to test the positive definiteness for multivariate forms. To solve this problem, Qi [1] provided that $f\left(x\right)$ as $\left(2\right)$ is positive definite, if and only if the real symmetric tensor $\mathcal{A}$ is positive definite, and presented an eigenvalue way to identify the positive definiteness of $\mathcal{A}$ with even m (see Proposition 1).

Proposition 1

([1]).Let $\mathcal{A}\in R^{[m,n]}$ be even-order symmetric. Then $\mathcal{A}$ is positive definite if and only if its H-eigenvalues are all positive.

However, it is not easy to compute all H-eigenvalues of $\mathcal{A}$ when m and n are large. Recently, Li et al. [11] gave a sufficient condition of the positive definiteness for an even-order symmetric tensor by $\mathcal{H}$-tensors (see Proposition 2). Here, a tensor $\mathcal{A}$ is called an $\mathcal{H}$-tensor [12] if there is a positive vector $x={(x_1,x_2,\dots ,x_n)}^T\in R^n$ satisfying

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

Proposition 2

([11]).Let $\mathcal{A}=\left(a_{i_1{i}_2\cdots i_m}\right)\in R^{[m,n]}$ be even-order symmetric with positive diagonal entries. If $\mathcal{A}$ is an $\mathcal{H}$-tensor, then $\mathcal{A}$ is positive definite.

Subsequently, based on the notion of diagonally dominant tensors, various criterions for $\mathcal{H}$-tensors are established [13,14,15,16,17,18,19,20,21], which only depend on the elements of the tensors, and are very effective for testing whether a tensor is an $\mathcal{H}$-tensor or not. For example,

Theorem 1

([17]).Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$. If

$$\begin{array}{c}\hfill |a_{ii\cdots i}|>\sum _{\genfrac{}{}{0pt}{}{i_2,i_3,\dots ,i_m\in N^{m-1}\backslash N_1^{m-1}}{{\delta }_{i{i}_2\cdots i_m}=0}}|a_{i{i}_2\cdots i_m}|+\sum _{i_2{i}_3\cdots i_m\in N_1^{m-1}}\underset{j\in \{i_2,i_3,\dots ,i_m\}}{max}\frac{R_j\left(\mathcal{A}\right)}{|a_{jj\cdots j}|}|a_{i{i}_2\cdots i_m}|,\forall i\in N_2\left(\mathcal{A}\right),\end{array}$$

where

$$N_1\left(\mathcal{A}\right)=\{i\in N:|a_{ii\cdots i}|>R_i\left(\mathcal{A}\right)\},N_2\left(\mathcal{A}\right)=\{i\in N:|a_{ii\cdots i}|\le R_i\left(\mathcal{A}\right)\},$$

then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Theorem 2

([18]).Let $\mathcal{A}=\left(a_{j_1\cdots j_m}\right)\in C^{[m,n]}$. If

$$\begin{array}{ccc}\hfill |a_{jj\cdots j}|r_j& >& \sum _{j_2{j}_3\cdots j_m\in N_0^{m-1}}|a_{j{j}_2\cdots j_m}|+\sum _{\genfrac{}{}{0pt}{}{j_2{j}_3\cdots j_m\in N_2^{m-1}}{{\delta }_{j{j}_2\cdots j_m}=0}}\underset{t\in \{j_2,j_3,\dots ,j_m\}}{max}\left\{r_t\right\}|a_{j{j}_2\cdots j_m}|\hfill \\ & & +\sum _{j_2{j}_3\cdots j_m\in N_1^{m-1}}\underset{t\in \{j_2,j_3,\cdots ,j_m\}}{max}\left\{l_t\right\}|a_{j{j}_2\cdots j_m}|,\forall j\in N_2\left(\mathcal{A}\right),\hfill \end{array}$$

and

$$|a_{ii\cdots i}|\ne \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_0^{m-1}}{{\delta }_{i{i}_2\cdots i_m}=0}}|a_{i{i}_2\cdots i_m}|,\forall i\in N_1\left(\mathcal{A}\right)\ne \varnothing (or{N}_1\left(\mathcal{A}\right)=\varnothing ),$$

where

$$\begin{array}{ccc}\hfill N_1\left(\mathcal{A}\right)& =& \{i\in N:0<|a_{ii\cdots i}|=R_i\left(\mathcal{A}\right)\},N_2\left(\mathcal{A}\right)=\{i\in N:0<|a_{ii\cdots i}|<R_i\left(\mathcal{A}\right)\},\hfill \\ \hfill N_3\left(\mathcal{A}\right)& =& \{i\in N:|a_{ii\cdots i}|>R_i\left(\mathcal{A}\right)\},\hfill \end{array}$$

then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Theorem 3

([21]).Let $\mathcal{A}=\left(a_{j_1\cdots j_m}\right)\in C^{[m,n]}$. If

$$\begin{array}{ccc}\hfill |a_{ii\cdots i}|& >& \frac{R_i\left(\mathcal{A}\right)}{R_i\left(\mathcal{A}\right)-|a_{ii\cdots i}|}\left[q\left(\sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}|a_{i{i}_2\cdots i_m}|+\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}}{{\delta }_{i{i}_2\cdots i_m}=0}}|a_{i{i}_2\cdots i_m}|\right)\right.\hfill \\ & & +\left.\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \{i_2,i_3,\dots ,i_m\}}{max}\frac{t{P}_j\left(\mathcal{A}\right)}{|a_{jj\cdots j}|}|a_{i{i}_2\cdots i_m}|\right],\forall i\in N_2\left(\mathcal{A}\right),\hfill \end{array}$$

and for $i\in N_1\left(\mathcal{A}\right)$, $|a_{ii\cdots i}|\ne {\displaystyle \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_0^{m-1}}{{\delta }_{i{i}_2\cdots i_m}=0}}}|a_{i{i}_2\cdots i_m}|$, where $N_1\left(\mathcal{A}\right)$, $N_2\left(\mathcal{A}\right)$, and $N_3\left(\mathcal{A}\right)$ are that as in Theorem 2, then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

We know that these criterions can be used just for a finite class of $\mathcal{H}$-tensors; that is, there are a lot of $\mathcal{H}$-tensors that cannot be identified by these existing criteria, and therefore, many authors consider whether there exists a more effective iterative scheme for testing $\mathcal{H}$-tensors [13,14,15,16,17,18,19,20]. However, some algorithms need to use a parameter $ϵ$, and it is hard to choose a suitable value for $ϵ$, and an inappropriate $ϵ$ may result in a large computing quantity. In this paper, we continue to propose some new sufficient conditions of $\mathcal{H}$-tensors, and we present a new non-parameter-involved iterative algorithm for testing $\mathcal{H}$-tensors. The obtained results extend the corresponding results in [17,18,19,20,21]. The validity of new methods is theoretically guaranteed, and the numerical experiments show their efficiency.

The remainder of this paper is organized as follows. In Section 2, we recall some related definitions, lemmas, and notations, which will be used in the proof of this paper. In Section 3, we explore some new sufficient conditions for identifying $\mathcal{H}$-tensors, only relying on the elements of such tensors. In Section 4, we further propose a non-parameter implementable iterative algorithm for identifying $\mathcal{H}$-tensors; the efficiency of the scheme is theoretically guaranteed in this section. In Section 5, as an application, we provide new conditions for testing the positive definiteness of even-order homogeneous multivariate forms. Some conclusions are summarized in Section 6.

2. Preliminaries

Now, some definitions, lemmas, and notations are recalled, which will be used in the sequel.

Definition 1

([4]).Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$. $\mathcal{A}$ is called reducible, if there exists a non-empty proper index subset $I\subset N$, such that

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

We call $\mathcal{A}$ irreducible if $\mathcal{A}$ is not reducible.

Definition 2

([2]).Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$ and $X=diag(x_1,x_2,\dots ,x_n)$. Denote

$$\begin{array}{c}\hfill \mathcal{B}=\left(b_{i_1\cdots i_m}\right)=\mathcal{A}{X}^{m-1},b_{i_1{i}_2\cdots i_m}=a_{i_1{i}_2\cdots i_m}{x}_{i_2}{x}_{i_3}\cdots x_{i_m},i_j\in N,j\in N.\end{array}$$

We call $\mathcal{B}$ the product of the tensor $\mathcal{A}$ and the matrix X.

Definition 3

([13]).Let $\mathcal{A}=\left(a_{i_1{i}_2\cdots i_m}\right)\in C^{[m,n]}$. For $i,j\in N(i\ne j)$, if there exist indices $k_1,k_2,\dots ,k_r$ with

$$\begin{array}{c}\hfill \sum _{\genfrac{}{}{0pt}{}{i_2,\dots ,i_m\in N}{{\delta }_{k_s{i}_2\cdots i_m}=0,k_{s+1}\in \{i_2,\dots ,i_m\}}}\left|a_{k_s{i}_2\cdots i_m}\right|\ne 0,s=0,1,\dots ,r,\end{array}$$

where $k_0=i, k_{r+1}=j$, then it is called that there is a non-zero elements chain from i to j.

Lemma 1

([11]).If $\mathcal{A}\in C^{[m,n]}$ is strictly diagonally dominant, then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Lemma 2

([2]).Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$. If there exists a positive diagonal matrix X, such that $\mathcal{A}{X}^{m-1}$ is an $\mathcal{H}$-tensor, then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Lemma 3

([11]).Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$. If $\mathcal{A}$ is irreducible,

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

and strict inequality holds for at least one i, then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Lemma 4

([13]).Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$. If

(i)

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

(ii)

$N_1\left(\mathcal{A}\right)=\left\{i\in N:\left|a_{ii\cdots i}\right|>R_i\left(\mathcal{A}\right)\right\}\ne \varnothing$,

(iii)

For any $i\notin N_1\left(\mathcal{A}\right)$, there exists a non-zero elements chain form i to j, such that $j\in N_1\left(\mathcal{A}\right)$, then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

For convenience, some notations that will be used in the following are given below. Let S be a non-empty subset of N, and let $N\backslash S$ be the complement of S in N. Given a tensor $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$ with $\left|a_{ii\cdots i}\right|\ne 0$, $\forall k\in Z^{+}=\left\{0,1,2,\cdots \right\}$, and

$$\begin{array}{ccc}\hfill R_i\left(\mathcal{A}\right)& =& \sum _{\genfrac{}{}{0pt}{}{i_2,\dots ,i_m\in N}{{\delta }_{i{i}_2\cdots i_m}=0}}|a_{i{i}_2\cdots i_m}|=\sum _{i_2,\dots ,i_m\in N}|a_{i{i}_2\cdots i_m}|-|a_{ii\cdots i}|,\hfill \\ \hfill N_1\left(\mathcal{A}\right)& =& \{i\in N:|a_{ii\cdots i}|=R_i\left(\mathcal{A}\right)\},\hfill \\ \hfill N_2\left(\mathcal{A}\right)& =& \{i\in N:|a_{ii\cdots i}|<R_i\left(\mathcal{A}\right)\},\hfill \\ \hfill N_3\left(\mathcal{A}\right)& =& \{i\in N:|a_{ii\cdots i}|>R_i\left(\mathcal{A}\right)\},\hfill \\ \hfill N^{m-1}\backslash S^{m-1}& =& \left\{i_2{i}_3\cdots i_m:i_2{i}_3\cdots i_m\in N^{m-1}and{i}_2{i}_3\cdots i_m\notin S^{m-1}\right\},\hfill \\ \hfill N_0^{m-1}& =& N^{m-1}\backslash \left(N_2^{m-1}\cup N_3^{m-1}\right),\hfill \\ \hfill y_i& =& \frac{\left|a_{ii\cdots i}\right|}{R_i\left(\mathcal{A}\right)},m=\underset{i\in N_2\left(\mathcal{A}\right)}{max}\frac{\left|a_{ii\cdots i}\right|}{R_i\left(\mathcal{A}\right)},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\omega }_0& =& \underset{i\in N_3\left(\mathcal{A}\right)}{max}\frac{\sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_3^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\left|a_{i{i}_2\cdots i_m}\right|}{\left|a_{ii\cdots i}\right|},\hfill \\ \hfill {\omega }_{k+1}& =& \underset{i\in N_3\left(\mathcal{A}\right)}{max}\frac{\sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+{\omega }_k\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_3^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\left|a_{i{i}_2\cdots i_m}\right|}{\left|a_{ii\cdots i}\right|},\hfill \\ \hfill {\theta }_{k+1,i}& =& \frac{\sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+{\omega }_k\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_3^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\left|a_{i{i}_2\cdots i_m}\right|}{\left|a_{ii\cdots i}\right|},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \delta & =& max\left\{{\omega }_{k+1},m\right\},\hfill \\ \hfill P_{k+1,i}\left(\mathcal{A}\right)& =& {\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|+{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|+{\omega }_k{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_3^{m-1}}}}\left|a_{i{i}_2\cdots i_m}\right|,\hfill \\ \hfill h_{k+1}& =& \frac{{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta {\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|}{P_{k+1,i}\left(\mathcal{A}\right)-{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_3^{m-1}}}}\underset{j\in \left\{i_2,i_2,\dots ,i_m\right\}}{max}\frac{P_{k+1,j}\left(\mathcal{A}\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}.\hfill \end{array}$$

3. Criteria for Identifying $\mathcal{H}$-Tensors

In this section, we give some new criteria for identifying $\mathcal{H}$-tensors.

Theorem 4.

Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$. If there exists $k\in Z^{+}$, such that

$$\begin{array}{ccc}\hfill \left|a_{ii\cdots i}\right|y_i& >& \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +h_{k+1}\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|,\forall i\in N_2\left(\mathcal{A}\right),\hfill \end{array}$$

(3)

and there exists $i_2{i}_3\cdots i_m\in N_3^{m-1}$ for any $i\in N_1\left(\mathcal{A}\right)$, such that $a_{i{i}_2{i}_3\cdots i_m}\ne 0$ $(or{N}_1\left(\mathcal{A}\right)=\varnothing )$, then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Proof. 

According to the definitions of ${\omega }_{k+1}$, $P_{k+1,i}\left(\mathcal{A}\right)$, by $0<{\omega }_{k+1}\le {\omega }_k<1$ and $0<m<1,$ for $\forall i\in N_3\left(\mathcal{A}\right)$, we have

$$\begin{array}{c}\hfill {\omega }_{k+1}\left|a_{ii\cdots i}\right|\ge \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+{\omega }_k\sum _{\underset{{\delta }_{i{i}_2\cdots i_m}=0}{i_2{i}_3\cdots i_m\in N_3^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|,\end{array}$$

so, $P_{k+1,i}\left(\mathcal{A}\right)\le {\omega }_{k+1}\left|a_{ii\cdots i}\right|\left(\forall i\in N_3\left(\mathcal{A}\right)\right)$, and

$$0<\frac{P_{k+1,i}\left(\mathcal{A}\right)}{\left|a_{ii\cdots i}\right|}\le {\omega }_{k+1}\le \delta <1,\forall i\in N_3\left(\mathcal{A}\right).$$

From the definition of $h_{k+1}$, we obtain:

$$\begin{array}{ccc}& & \frac{{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta {\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|}{P_{k+1,i}\left(\mathcal{A}\right)-{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_1^{m-1}}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\frac{P_{k+1,j}\left(\mathcal{A}\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}\hfill \\ & \le & \frac{\delta \left({\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|+{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|\right)}{P_{k+1,i}\left(\mathcal{A}\right)-{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_3^{m-1}}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\frac{P_{k+1,j}\left(\mathcal{A}\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}\hfill \\ & =& \frac{\delta \left(P_{k+1,i}\left(\mathcal{A}\right)-{\omega }_k{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_1^{m-1}}}}\left|a_{i{i}_2\cdots i_m}\right|\right)}{P_{k+1,i}\left(\mathcal{A}\right)-{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_3^{m-1}}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\frac{P_{k+1,j}\left(\mathcal{A}\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}\hfill \\ & \le & \frac{\delta \left(P_{k+1,i}\left(\mathcal{A}\right)-{\omega }_{k+1}{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_1^{m-1}}}}\left|a_{i{i}_2\cdots i_m}\right|\right)}{P_{k+1,i}\left(\mathcal{A}\right)-{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_1^{m-1}}}}{\displaystyle \underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}}\frac{P_{k+1,j}\left(\mathcal{A}\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}\hfill \\ & =& \frac{\delta \left(P_{k+1,i}\left(\mathcal{A}\right)-{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_3^{m-1}}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\frac{P_{k+1,j}\left(\mathcal{A}\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|\right)}{P_{k+1,i}\left(\mathcal{A}\right)-{\displaystyle \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_3^{m-1}}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\frac{P_{k+1,j}\left(\mathcal{A}\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|}\hfill \\ & <& 1,\hfill \end{array}$$

so $0\le h_{k+1}<1$, and

$$\begin{array}{ccc}\hfill h_{k+1}{P}_{k+1,i}\left(\mathcal{A}\right)& \ge & \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +h_{k+1}\sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_3^{m-1}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\frac{P_{k+1,j}\left(\mathcal{A}\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|.\hfill \end{array}$$

(4)

For any $i\in N_2\left(\mathcal{A}\right)$, let

$$\begin{array}{c}{\varphi }_i(\mathcal{A})=\frac{1}{{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|}(\left|a_{ii\cdots i}\right|y_i-{\displaystyle \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ -\delta {\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|-h_{k+1}{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|).\hfill \end{array}$$

(5)

When ${\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_1^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|=0$, denote ${\varphi }_i\left(\mathcal{A}\right)=+\infty$. By (3) and (5), we have ${\varphi }_i\left(\mathcal{A}\right)>0\left(\forall i\in N_2\left(\mathcal{A}\right)\right)$. Since $0\le \underset{i\in N_3\left(\mathcal{A}\right)}{max}\left(h_{k+1}{\theta }_{k+1,i}\right)<\delta$, then there is a sufficiently small positive number $\epsilon$, such that

$$0<\epsilon <\underset{i\in N_2\left(\mathcal{A}\right)}{min}{\varphi }_i\left(\mathcal{A}\right)\le +\infty ,0<\underset{i\in N_3\left(\mathcal{A}\right)}{max}\left(\epsilon +h_{k+1}{\theta }_{k+1,i}\right)<\delta <1.$$

Let diagonal matrix $X=diag(x_1,x_2,\dots ,x_n)$, where

$$\begin{array}{c}\hfill x_i=\left\{\begin{array}{cc}\hfill {\delta }^{\frac{1}{m-1}},\hfill & i\in N_1\left(\mathcal{A}\right),\hfill \\ \hfill {y_i}^{\frac{1}{m-1}},\hfill & i\in N_2\left(\mathcal{A}\right),\hfill \\ \hfill {\left(h_{k+1}{\theta }_{k+1,i}+\epsilon \right)}^{\frac{1}{m-1}},\hfill & i\in N_3\left(\mathcal{A}\right).\hfill \end{array}\right.\end{array}$$

As $\epsilon \ne +\infty$, $x_i\ne +\infty$, which implies that X is a diagonal matrix with positive entries. Let $\mathcal{B}=\left(b_{i_1{i}_2\cdots i_m}\right)=\mathcal{A}{X}^{m-1}$. Next, we will prove that $\mathcal{B}$ is a strictly diagonally dominant tensor.

(1) For any $i\in N_1\left(\mathcal{A}\right)$, since

$$0<\underset{i\in N_2\left(\mathcal{A}\right)}{max}\left\{y_i\right\}\le \delta <1,0<\underset{i\in N_3\left(\mathcal{A}\right)}{max}\left\{h_{k+1}{\theta }_{k+1,i}+\epsilon \right\}<\delta <1,$$

then

$$\begin{array}{ccc}\hfill R_i\left(\mathcal{B}\right)& =& \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|{\left(y_{i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(y_{i_m}\right)}^{\frac{1}{m-1}}+\sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}\cdots x_{i_m}\hfill \\ & & +\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|{\left(h_{k+1}{\theta }_{k+1,i_2}+\epsilon \right)}^{\frac{1}{m-1}}\cdots {\left(h_{k+1}{{\theta }_{k+1,}}_{i_m}+\epsilon \right)}^{\frac{1}{m-1}}\hfill \\ & \le & \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{\underset{{\delta }_{i{i}_2\cdots i_m}=0}{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{h_{k+1}{\theta }_{k+1,j}+\epsilon \right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & <& \delta \left(\sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{\underset{{\delta }_{i{i}_2\cdots i_m}=0}{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\right)\hfill \\ & =& \delta \left|a_{ii\cdots i}\right|=\left|b_{ii\cdots i}\right|.\hfill \end{array}$$

(6)

(2) For any $i\in N_2\left(\mathcal{A}\right)$, if ${\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|+{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_1^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|=0$, we have

$$\begin{array}{ccc}\hfill R_i\left(\mathcal{B}\right)& =& \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_2^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|{\left(y_{i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(y_{i_m}\right)}^{\frac{1}{m-1}}\le \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_2^{m-1}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & <& y_i\left|a_{ii\cdots i}\right|=\left|b_{ii\cdots i}\right|.\hfill \end{array}$$

(7)

If ${\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|+{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_1^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|\ne 0$, by Equality (5), we get

$$\begin{array}{ccc}\hfill R_i\left(\mathcal{B}\right)& =& \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\left|a_{i{i}_2\cdots i_m}\right|{\left(y_{i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(y_{i_m}\right)}^{\frac{1}{m-1}}+\sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}\cdots x_{im}\hfill \\ & & +\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|{\left(h_{k+1}{\theta }_{k+1,i_2}+\epsilon \right)}^{\frac{1}{m-1}}\cdots {\left(h_{k+1}{{\theta }_{k+1,}}_{i_m}+\epsilon \right)}^{\frac{1}{m-1}}\hfill \\ & \le & \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{h_{k+1}{\theta }_{k+1,j}+\epsilon \right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & =& \epsilon \sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+h_{k+1}\sum _{i_2{i}_3\cdots i_m\in N_1^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & <& {\varphi }_i\left(\mathcal{A}\right)\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+h_{k+1}\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & =& y_i\left|a_{ii\cdots i}\right|=\left|b_{ii\cdots i}\right|.\hfill \end{array}$$

(8)

(3) For any $i\in N_3\left(\mathcal{A}\right)$, by Inequality (4), it holds that

$$\begin{array}{ccc}\hfill R_i\left(\mathcal{B}\right)& =& \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|{\left(y_{i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(y_{i_m}\right)}^{\frac{1}{m-1}}+\sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}\cdots x_{im}\hfill \\ & & +\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_3^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\left|a_{i{i}_2\cdots i_m}\right|{\left(h_{k+1}{\theta }_{k+1,i_2}+\epsilon \right)}^{\frac{1}{m-1}}\cdots {\left(h_{k+1}{{\theta }_{k+1,}}_{i_m}+\epsilon \right)}^{\frac{1}{m-1}}\hfill \\ & \le & \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_3^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{h_{k+1}{\theta }_{k+1,j}+\epsilon \right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & =& \epsilon \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_3^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+h_{k+1}\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_1^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\frac{P_{k+1,j}\left(A\right)}{\left|a_{jj\cdots j}\right|}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & \le & \epsilon \sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+h_{k+1}{P}_{k+1,i}\left(\mathcal{A}\right)\hfill \\ & <& \epsilon \left|a_{ii\cdots i}\right|+h_{k+1}{P}_{k+1,i}\left(\mathcal{A}\right)=\left|b_{ii\cdots i}\right|.\hfill \end{array}$$

(9)

Hence, by Inequalities (6)–(9), it holds that $|b_{ii\cdots i}|>R_i\left(\mathcal{B}\right)(\forall i\in N)$; that is, $\mathcal{B}$ is a strictly diagonally dominant tensor. From Lemma 1, $\mathcal{B}$ is an $\mathcal{H}$-tensor. By Lemma 2, $\mathcal{A}$ is an $\mathcal{H}$-tensor. □

Theorem 5.

Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$ be irreducible. If there exists $k\in Z^{+}$ such that

$$\begin{array}{ccc}\hfill \left|a_{ii\cdots i}\right|y_i& \ge & \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +h_{k+1}\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|,\forall i\in N_2\left(\mathcal{A}\right),\hfill \end{array}$$

(10)

and at least one strict inequality in $\left(10\right)$ holds, then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Proof. 

Let diagonal matrix $X=diag(x_1,x_2,\dots ,x_n)$, where

$$x_i=\left\{\begin{array}{cc}\hfill {\delta }^{\frac{1}{m-1}},\hfill & i\in N_1\left(\mathcal{A}\right),\hfill \\ \hfill {y_i}^{\frac{1}{m-1}},\hfill & i\in N_2\left(\mathcal{A}\right),\hfill \\ \hfill {\left(h_{k+1}{\theta }_{k+1,i}\right)}^{\frac{1}{m-1}},\hfill & i\in N_3\left(\mathcal{A}\right).\hfill \end{array}\right.$$

This shows that X is a diagonal matrix with positive entries. Let $\mathcal{B}=\left(b_{i_1{i}_2\cdots i_m}\right)=\mathcal{A}{X}^{m-1}$. Next, we prove that $\mathcal{B}$ is a diagonally dominant tensor.

(1) For $\forall i\in N_1\left(\mathcal{A}\right)$, since

$$0<\underset{i\in N_2\left(\mathcal{A}\right)}{max}\left\{y_i\right\}\le \delta <1,0\le \underset{i\in N_3\left(\mathcal{A}\right)}{max}\left\{h_{k+1}{\theta }_{k+1,i}\right\}<\delta <1,$$

then

$$\begin{array}{ccc}\hfill R_i\left(\mathcal{B}\right)& =& \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|{\left(y_{i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(y_{i_m}\right)}^{\frac{1}{m-1}}+\sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}\cdots x_{i_m}\hfill \\ & & +\sum _{i_2{i}_3\cdots i_m\in N_1^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|{\left(h_{k+1}{\theta }_{k+1,i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(h_{k+1}{{\theta }_{k+1,}}_{i_m}\right)}^{\frac{1}{m-1}}\hfill \\ & \le & \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{\underset{{\delta }_{i{i}_2\cdots i_m}=0}{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +h_{k+1}\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & \le & \delta \left(\sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{\underset{{\delta }_{i{i}_2\cdots i_m}=0}{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|+\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\right)\hfill \\ & =& \delta \left|a_{ii\cdots i}\right|=\left|b_{ii\cdots i}\right|.\hfill \end{array}$$

(11)

(2) For $\forall i\in N_2\left(\mathcal{A}\right)$, if ${\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|+{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|=0$, we obtain

$$\begin{array}{ccc}\hfill R_i\left(\mathcal{B}\right)& =& \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_2^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|{\left(y_{i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(y_{i_m}\right)}^{\frac{1}{m-1}}\le \sum _{\underset{{\delta }_{i{i}_2\cdots i_m=0}}{i_2{i}_3\cdots i_m\in N_2^{m-1}}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & \le & y_i\left|a_{ii\cdots i}\right|=\left|b_{ii\cdots i}\right|.\hfill \end{array}$$

(12)

If ${\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|+{\displaystyle \sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}}\left|a_{i{i}_2\cdots i_m}\right|\ne 0$, by Inequality (10), we get

$$\begin{array}{ccc}\hfill R_i\left(\mathcal{B}\right)& =& \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\left|a_{i{i}_2\cdots i_m}\right|{\left(y_{i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(y_{i_m}\right)}^{\frac{1}{m-1}}+\sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}\cdots x_{im}\hfill \\ & & +\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|{\left(h_{k+1}{\theta }_{k+1,i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(h_{k+1}{{\theta }_{k+1,}}_{i_m}\right)}^{\frac{1}{m-1}}\hfill \\ & \le & \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +h_{k+1}\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & \le & y_i\left|a_{ii\cdots i}\right|=\left|b_{ii\cdots i}\right|.\hfill \end{array}$$

(13)

(3) For $\forall i\in N_1\left(\mathcal{A}\right)$, by Inequality (4), it holds that

$$\begin{array}{ccc}\hfill R_i\left(\mathcal{B}\right)& =& \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|{\left(y_{i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(y_{i_m}\right)}^{\frac{1}{m-1}}+\sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|x_{i_2}\cdots x_{im}\hfill \\ & & +\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_3^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\left|a_{i{i}_2\cdots i_m}\right|{\left(h_{k+1}{\theta }_{k+1,i_2}\right)}^{\frac{1}{m-1}}\cdots {\left(h_{k+1}{{\theta }_{k+1,}}_{i_m}\right)}^{\frac{1}{m-1}}\hfill \\ & \le & \sum _{i_2{i}_3\cdots i_m\in N_2^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +h_{k+1}\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_3^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & \le & h_{k+1}{P}_{k+1,j}\left(\mathcal{A}\right)=\left|b_{ii\cdots i}\right|.\hfill \end{array}$$

(14)

Therefore, by Inequalities (11)–(14), we conclude that $|b_{ii\cdots i}|\ge R_i\left(\mathcal{B}\right)$ for all $i\in N$, and for all $i\in N_2\left(\mathcal{B}\right)$, at least one strict inequality in (12) and (13) holds; that is, there exists an $i_0\in N_2\left(\mathcal{B}\right)$ satisfying $|b_{i_0{i}_0\cdots i_0}|>R_{i_0}\left(\mathcal{B}\right)$.

Meanwhile, $\mathcal{A}$ is irreducible, and so is $\mathcal{B}$. By Lemma 3, we obtain that $\mathcal{B}$ is an $\mathcal{H}$-tensor. From Lemma 2, $\mathcal{A}$ is also an $\mathcal{H}$-tensor. □

Let

$$\begin{array}{ccc}\hfill J\left(\mathcal{A}\right)& =& \left\{i\in N_2\left(\mathcal{A}\right):|a_{ii\cdots i}|y_i>\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_i\right\}\left|a_{i{i}_2\cdots i_m}\right|\right.\hfill \\ & & +\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}|a_{i{i}_2\cdots i_m}|+h_{k+1}\left.\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|\right\}.\hfill \end{array}$$

Theorem 6.

Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$. For any $i\in N_2\left(\mathcal{A}\right)$, there exists $k\in Z^{+}$ such that

$$\begin{array}{ccc}\hfill \left|a_{ii\cdots i}\right|y_i& \ge & \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^{m-1}\hfill }{{\delta }_{i{i}_2\cdots i_m}=0\hfill }}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{y_j\right\}\left|a_{i{i}_2\cdots i_m}\right|+\delta \sum _{i_2{i}_3\cdots i_m\in N_0^{m-1}}\left|a_{i{i}_2\cdots i_m}\right|\hfill \\ & & +h_{k+1}\sum _{i_2{i}_3\cdots i_m\in N_3^{m-1}}\underset{j\in \left\{i_2,i_3,\dots ,i_m\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{i{i}_2\cdots i_m}\right|,\hfill \end{array}$$

(15)

and for $\forall i\in N\backslash J\left(\mathcal{A}\right)\ne \varnothing$, there is a non-zero elements chain from i to j satisfying $j\in J\left(\mathcal{A}\right)\ne \varnothing$, then $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Proof. 

Denote diagonal matrix $X=diag(x_1,x_2,\dots ,x_n)$ as that in the proof of Theorem 5. This shows that X is a diagonal matrix with positive entries. Mark $\mathcal{B}=\left(b_{i_1{i}_2\cdots i_m}\right)=\mathcal{A}{X}^{m-1}$. Similar to the process in proof of Theorem 4, it holds that $\left|b_{ii\cdots i}\right|\ge R_i\left(\mathcal{B}\right)(\forall i\in N)$, and there is at least one $\left|b_{ii\cdots i}\right|>R_i\left(\mathcal{B}\right)$.

On the other hand, if $\left|b_{ii\cdots i}\right|=R_i\left(\mathcal{B}\right)$, then $\forall i\in N\backslash J\left(\mathcal{A}\right)$. We know that there exists a chain of non-zero elements from i to j in $\mathcal{A}$, with $j\in J\left(\mathcal{A}\right)$. So, there is also a non-zero element chain from i to j in $\mathcal{B}$ satisfying $\left|b_{jj\cdots j}\right|>R_j\left(\mathcal{B}\right)$. Hence, $\mathcal{B}$ satisfies the conditions of Lemma 4; that is, $\mathcal{B}$ is an $\mathcal{H}$-tensor. From Lemma 2, $\mathcal{A}$ is also an $\mathcal{H}$-tensor. □

Example 1.

Given tensor $\mathcal{A}=\left(a_{i_1{i}_2{i}_3}\right)\in C^{[3,3]}$, defined as follows:

$$\mathcal{A}=\left[A\right(1,:,:),A(2,:,:),A(3,:,:\left)\right],$$

$$A(1,:,:)=\left(\begin{array}{ccc}12& 3& 1\\ 2& 4& 0\\ 2& 0& 6\end{array}\right),A(2,:,:)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 6& 0\\ 0& 0& 3\end{array}\right),A(3,:,:)=\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 20\end{array}\right).$$

By calculations, we have

$$|a_{111}|=12,R_1\left(\mathcal{A}\right)=18,|a_{222}|=6,R_2\left(\mathcal{A}\right)=4,|a_{333}|=20,R_3\left(\mathcal{A}\right)=3.$$

Then $N_3\left(\mathcal{A}\right)=\{2,3\}$, $N_2\left(\mathcal{A}\right)=\left\{1\right\}$, $N_1\left(\mathcal{A}\right)=\varnothing$, and

$$\begin{array}{ccc}& & y_1=\frac{2}{3},m=\frac{2}{3},{\omega }_0=\frac{2}{3},{\omega }_1=\frac{1}{2},{\omega }_2=\frac{5}{12}\hfill \\ & & {\theta }_{3,2}=\frac{9}{24},{\theta }_{3,3}=\frac{29}{240},P_{3,2}=\frac{9}{4},P_{3,3}=\frac{29}{20},h_3=\frac{32}{49}.\hfill \end{array}$$

When $i=1$ and $k=2$, since

$$\begin{array}{ccc}& & \sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\in N_2^2\hfill }{{\delta }_{1i_2{i}_3}=0\hfill }}\underset{j\in \left\{i_2,i_3\right\}}{max}\left\{y_j\right\}\left|a_{1i_2{i}_3}\right|+\delta \sum _{i_2{i}_3\in N_0^2}\left|a_{1i_2{i}_3}\right|+h_{k+1}\sum _{i_2{i}_3\in N_3^2}\underset{j\in \left\{i_2,i_3\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{1i_2{i}_3}\right|\hfill \\ & =& 8\times \frac{2}{3}+\frac{32}{49}\times \frac{9}{24}\times 10=\frac{1144}{147}<8=\left|a_{111}\right|y_1,\hfill \end{array}$$

then $\mathcal{A}$ satisfies the conditions of Theorem 4; that is, $\mathcal{A}$ is an $\mathcal{H}$-tensor. However,

$$\begin{array}{c}\hfill \sum _{\underset{{\delta }_{1i_2{i}_3}=0}{i_2{i}_3\in N^2/N_3^2}}\left|a_{1i_2{i}_3}\right|+\sum _{i_2{i}_3\in N_3^2}\underset{j\in \{i_2,i_3\}}{max}\frac{R_j\left(\mathcal{A}\right)}{\left|a_{jjj}\right|}\left|a_{1i_2{i}_3}\right|=\frac{44}{3}>12=\left|a_{111}\right|,\end{array}$$

$$\begin{array}{c}\hfill \sum _{i_2{i}_3\in N_0^2}|a_{1i_2{i}_3}|+\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\in N_2^2}{{\delta }_{1i_2{i}_3}=0}}\underset{j\in \{i_2,i_3\}}{max}\left\{r_j\right\}|a_{1i_2{i}_3}|+\sum _{i_2{i}_3\in N_3^2}\underset{j\in \{i_2,i_3\}}{max}\left\{t_j\right\}|a_{1i_2{i}_3}|=\frac{82}{9}>4=|a_{111}|r_1,\end{array}$$

and

$$\begin{array}{ccc}& & \frac{R_1\left(\mathcal{A}\right)}{R_1\left(\mathcal{A}\right)-|a_{111}|}\left[q\left(\sum _{i_2{i}_3\cdots i_m\in N_0^2}|a_{1i_2{i}_3}|+\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3\cdots i_m\in N_2^2}{{\delta }_{1i_2{i}_3}=0}}|a_{1i_2{i}_3}|\right)+\sum _{i_2{i}_3\in N_3^2}\underset{j\in \{i_2,i_3\}}{max}\frac{t{P}_j\left(\mathcal{A}\right)}{|a_{jjj}|}|a_{1i_2{i}_3}|\right]\hfill \\ & =& \frac{18}{18-12}\left[\frac{1}{3}(2+2+3+1)+\frac{4}{21}(4+6)\right]=\frac{96}{7}>12=|a_{111}|.\hfill \end{array}$$

Therefore, $\mathcal{A}$ does not satisfy the conditions of Theorem 1 (Theorem 1 in [17]), Theorem 2 (Theorem 1 in [18]), and Theorem 3 (Theorem 2 in [21]), respectively.

Remark 1.

By Example 1, it is easy to see that Theorem 4 is better than the other ones, but it is difficult to say in advance which one is better. However, from Theorem 4, we conclude that $0\le \delta <1$, $0\le h_{k+1}\le 1$, and

$$\begin{array}{c}\hfill h_{k+1}{\theta }_{k+1,i}<\frac{R_i\left(\mathcal{A}\right)}{|a_{ii\cdots i}|}<1,\forall i\in N_3\left(\mathcal{A}\right).\end{array}$$

Thus, all conditions in Theorem 4 are weaker than those in Theorem 1.

4. An Iterative Algorithm for Testing $\mathcal{H}$-Tensors

In this section, we propose a new iterative scheme for testing $\mathcal{H}$-tensors based on the obtained results in Section 3.

The theoretical analysis of Algorithm 1 can be obtained from the following result.

Algorithm 1: A new iterative scheme for identifying $\mathcal{H}$-tensors.

INPUT: $\mathcal{A}=\left(a_{i_1{i}_2\cdots i_m}\right)\in C^{[m,n]}$ with $a_{ii\cdots i}\ne 0$, $\forall i\in N$.

OUTPUT: diagonal matrix $X=X^{\left(0\right)}{X}^{\left(1\right)}\cdots X^{(t-1)}$ with positive entries if $\mathcal{A}$ is an $\mathcal{H}$-tensor.

Step 1. If $N_3\left(\mathcal{A}\right)=\varnothing$, then $\mathcal{A}$ is not an $\mathcal{H}$-tensor, stop; otherwise,

Step 2. Set ${\mathcal{A}}^{\left(0\right)}=\mathcal{A}$, $X^{\left(0\right)}=\mathcal{I}$, $t=1$.

Step 3. Compute ${\mathcal{A}}^{\left(t\right)}={\mathcal{A}}^{(t-1)}{\left(X^{(t-1)}\right)}^{m-1}=\left(a_{i_1{i}_2\cdots i_m}^{\left(t\right)}\right)$.

Step 4. If $N_2\left({\mathcal{A}}^{\left(t\right)}\right)=\varnothing$, then $\mathcal{A}$ is an $\mathcal{H}$-tensor, stop; otherwise,

Step 5. If $N_1\left({\mathcal{A}}^{\left(t\right)}\right)=\varnothing$, then $\mathcal{A}$ is not an $\mathcal{H}$-tensor, stop; otherwise,

Step 6. Compute $y_i^{\left(t\right)}$, ${\delta }^{\left(t\right)}$, $h_{k+1}^{\left(t\right)}$, $P_{k+1,i}\left({\mathcal{A}}^{\left(t\right)}\right)$, $\forall k\in Z^{+}$, $\forall i\in N$.

Step 7. If there exists $k\in Z^{+}$ satisfying $$\begin{gathered} |a_{ii\cdots i}^{\left(t\right)}|y_i^{\left(t\right)}>{\delta }^{\left(t\right)}\sum _{i_2\cdots i_m\in N_0^{m-1}\left({\mathcal{A}}^{\left(t\right)}\right)}|a_{i{i}_2\cdots i_m}^{\left(t\right)}|+\sum _{\genfrac{}{}{0pt}{}{i_2\cdots i_m\in N_2^{m-1}\left({\mathcal{A}}^{\left(t\right)}\right)}{{\delta }_{i{i}_2\dots i_m}=0}}\underset{j\in \{i_2,i_3,\dots ,i_m\}}{max}\left\{y_j^{\left(t\right)}\right\}|a_{i{i}_2\cdots i_m}^{\left(t\right)}| \\ +h_{k+1}^{\left(t\right)}\sum _{i_2\cdots i_m\in N_3^{m-1}\left({\mathcal{A}}^{\left(t\right)}\right)}\underset{j\in \{i_2,i_3,\dots ,i_m\}}{max}\left\{\frac{P_{k+1,j}\left({\mathcal{A}}^{\left(t\right)}\right)}{|a_{jj\cdots j}^{\left(t\right)}|}\right\}|a_{i{i}_2\cdots i_m}^{\left(t\right)}|,\forall i\in N_2\left({\mathcal{A}}^{\left(t\right)}\right), \end{gathered}$$ and $$|a_{ii\cdots i}^{\left(t\right)}|\ne \sum _{\genfrac{}{}{0pt}{}{i_2\cdots i_m\in N_3^{m-1}\left({\mathcal{A}}^{\left(t\right)}\right)}{{\delta }_{i{i}_2\cdots i_m}=0}}|a_{i{i}_2\cdots i_m}^{\left(t\right)}|,\forall i\in N_1\left({\mathcal{A}}^{\left(t\right)}\right)\ne \varnothing (or{N}_1\left({\mathcal{A}}^{\left(t\right)}\right)=\varnothing ),$$ then $\mathcal{A}$ is an $\mathcal{H}$-tensor, stop; otherwise,

Step 8. Set $X^{\left(t\right)}=diag(x_1^{\left(t\right)},x_2^{\left(t\right)},\dots ,x_n^{\left(t\right)})$, compute $$x_i^{\left(t\right)}=\left\{\begin{array}{cc}{\left({\delta }^{\left(t\right)}\right)}^{\frac{1}{m-1}},& \forall i\in N_1\left({\mathcal{A}}^{\left(t\right)}\right),\\ {\left(y_i^{\left(t\right)}\right)}^{\frac{1}{m-1}},& \forall i\in N_2\left({\mathcal{A}}^{\left(t\right)}\right),\\ {\left(\frac{h_{k+1}^{\left(t\right)}{P}_{k+1,i}\left({\mathcal{A}}^{\left(t\right)}\right)}{|a_{ii\cdots i}^{\left(t\right)}|}\right)}^{\frac{1}{m-1}},& \forall i\in N_3\left({\mathcal{A}}^{\left(t\right)}\right).\end{array}\right.$$

Step 9. Set $X^{\left(t\right)}=diag(x_1^{\left(t\right)},x_2^{\left(t\right)},\dots ,x_n^{\left(t\right)})$, $t=t+1$; go to step 3.

Theorem 7.

Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$ be an $\mathcal{H}$-tensor. Then, Algorithm 1 terminates after a finite number of iterations by producing a strictly diagonally dominant tensor.

Proof. 

Without loss of generality, suppose that $\mathcal{A}$ is a non-negative tensor. Instead, the Algorithm 1 yields an infinite sequence $\left\{{\mathcal{A}}^{\left(t\right)}\right\}$, with

$${\mathcal{A}}^{\left(t\right)}={\mathcal{A}}^{\left(0\right)}{\left(X^{\left(1\right)}\right)}^{m-1}{\left(X^{\left(2\right)}\right)}^{m-1}\cdots {\left(X^{(t-1)}\right)}^{m-1}.$$

By the fact that each diagonal entry of matrix $X^{\left(t\right)}$ is less than 1, we have

$$\mathcal{A}={\mathcal{A}}^{\left(0\right)}={\mathcal{A}}^{\left(1\right)}\ge \cdots \ge {\mathcal{A}}^{\left(t\right)}\ge \cdots \ge 0.$$

Hence, the sequence $\left\{{\mathcal{A}}^{\left(t\right)}\right\}$ has a limitation,

$$\mathcal{B}=\underset{t\to +\infty }{lim}{\mathcal{A}}^{\left(t\right)}\ge 0,$$

where $\mathcal{B}=\mathcal{A}{X}^{m-1}$, $X=X^{\left(1\right)}{X}^{\left(2\right)}\cdots X^{\left(t\right)}\cdots$ is a diagonal matrix with positive entries.

Next, we will prove that

$$\underset{t\to +\infty }{lim}{N}_3\left({\mathcal{A}}^{\left(t\right)}\right)=N_3\left(\mathcal{B}\right)=\varnothing .$$

In fact, suppose that $\underset{t\to +\infty }{lim}{N}_3\left({\mathcal{A}}^{\left(t\right)}\right)\ne \varnothing$, then

$$1-{\delta }^{\left(t\right)}>0,1-y_i^{\left(t\right)}>0,1-\frac{h_{k+1}^{\left(t\right)}{P}_{k+1,i}\left({\mathcal{A}}^{\left(t\right)}\right)}{|a_{ii\cdots i}^{\left(t\right)}|}>0.$$

Therefore, there exists $i\in N_1\left({\mathcal{A}}^{\left(p\right)}\right)$ and positive numbers ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$, such that

$$a_{ii\cdots i}^{\left(t\right)}\left(1-{\delta }^{\left(t\right)}\right)>{\epsilon }_1,a_{ii\cdots i}^{\left(t\right)}\left(1-y_i^{\left(t\right)}\right)>{\epsilon }_2,a_{ii\cdots i}^{\left(t\right)}\left(1-\frac{h_{k+1}^{\left(t\right)}{P}_{k+1,i}\left({\mathcal{A}}^{\left(t\right)}\right)}{|a_{ii\cdots i}^{\left(t\right)}|}\right)>{\epsilon }_3.$$

Denote ${\epsilon }_0=min\{{\epsilon }_1,{\epsilon }_2,{\epsilon }_3\}$. Based on Algorithm 1, for $\forall i\in N$, it holds that

$$\begin{array}{ccc}& & 0<a_{ii\cdots i}^{(t+1)}=a_{ii\cdots i}^{\left(t\right)}{\delta }^{\left(t\right)}<a_{ii\cdots i}^{\left(t\right)}-{\epsilon }_1<a_{ii\cdots i}^{\left(t\right)}-{\epsilon }_0,\hfill \\ & & 0<a_{ii\cdots i}^{(t+1)}=a_{ii\cdots i}^{\left(t\right)}{y}_i^{\left(t\right)}<a_{ii\cdots i}^{\left(t\right)}-{\epsilon }_2<a_{ii\cdots i}^{\left(t\right)}-{\epsilon }_0,\hfill \\ & & 0<a_{ii\cdots i}^{(t+1)}=a_{ii\cdots i}^{\left(t\right)}\frac{h_{k+1}^{\left(t\right)}{P}_{k+1,i}\left({\mathcal{A}}^{\left(t\right)}\right)}{|a_{ii\cdots i}^{\left(t\right)}|}<a_{ii\cdots i}^{\left(t\right)}-{\epsilon }_3<a_{ii\cdots i}^{\left(t\right)}-{\epsilon }_0.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill a_{ii\cdots i}^{\left(0\right)}=a_{ii\cdots i}^{\left(1\right)}>a_{ii\cdots i}^{\left(2\right)}+{\epsilon }_0>\cdots >a_{ii\cdots i}^{\left(t\right)}+(t-1){\epsilon }_0.\end{array}$$

When $t\to +\infty$, one has $a_{ii\cdots i}^{\left(0\right)}\to +\infty$. A contradiction arrives, which means that

$$\underset{t\to +\infty }{lim}{N}_3\left({\mathcal{A}}^{\left(t\right)}\right)=N_3\left(\mathcal{B}\right)=\varnothing ,$$

that is,

$$|b_{ii\cdots i}|\le R_i\left(\mathcal{B}\right),\forall i\in N.$$

Hence, $\mathcal{B}$ is not an $\mathcal{H}$-tensor [11]. Meanwhile, if $\mathcal{A}$ is an $\mathcal{H}$-tensor, then there is a diagonal matrix D with positive entries satisfying $\mathcal{A}{D}^{m-1}=\mathcal{B}{\left(X^{-1}D\right)}^{m-1}$ is a strictly diagonally dominant tensor. Hence, $\mathcal{B}$ is an $\mathcal{H}$-tensor, which is a contradiction. Therefore, Algorithm 1 stops after a finite steps of iterations. □

Example 2

([17]).Randomly generate 100 mth-order n-dimensional tensors, such that the elements of each tensor satisfy

$$a_{i_1{i}_2\cdots i_m}\in \left\{\begin{array}{cc}(-n^m\times 0.6,n^m\times 0.6),& if{i}_1=\cdots =i_m,\\ (-1,1),& otherwise.\end{array}\right.$$

The numerical results of Algorithm 1 are shown in

Table 1. Numerical results of Example 2.

m (Order)	n (Dimension)	${\mathit{t}}_1$	${\mathit{t}}_2$	${\mathit{t}}_3$	CPU Time (s)
4	6	83	0	17	57.3690
4	10	78	0	22	55.8954
4	15	73	0	27	60.4535
4	20	67	0	33	84.5546
4	40	81	0	19	120.3584
5	6	73	0	27	46.5728
5	10	82	0	18	50.4879
5	15	85	0	15	235.0412
5	20	64	0	36	300.2586
5	40	75	0	25	276.8509
6	6	93	0	7	285.4570
6	10	85	0	15	254.2385
6	15	67	0	33	106.8336
6	20	78	0	22	331.1455
6	40	81	0	19	298.4535

, where $t_1$, $t_2$, and $t_3$ denote the number of tensors that the input tensor is, or is not an $\mathcal{H}$-tensor, and undetermined, within a certain number of iterations, respectively.

Example 3

([14]).Randomly generate fourth-order 10-dimensional tensors satisfying that the elements of each tensor are from $[-1,1]$. For each tensor, the following modifications are satisfied: Each tensor can be symmetrized, and all diagonal entries can be replaced with absolute value; all diagonal entries of each tensor can be amplified to a degree such that some of these tensors are positive definite.

The numerical results of Algorithm 1 are shown in

Table 2. Numerical results of Example 3.

Amplification	Iteration Steps	Positive Definiteness	CPU Time (s)
500	1	N	$0.04358$
1000	10	Y	$0.05870$
1000	19	−	$0.09405$
1000	3	N	$0.06088$
1500	2	Y	$0.05203$
1500	5	Y	$0.05019$
1500	8	N	$0.05976$
1500	3	Y	$0.06257$
1800	1	Y	$0.05250$
2000	2	Y	$0.05147$
2000	1	Y	$0.05055$

, where "Y", "N", and symbol "-" denote the output result that the input tensor is or is not positive definite and undetermined within a certain number of iterations, respectively.

According to Table 1 and Table 2, we easily observe that Algorithm 1 is much more efficient for identifying $\mathcal{H}$-tensors and the positive definiteness of the input tensor in most cases.

5. An Application: The Positive Definiteness of Homogeneous Polynomial Forms

In this section, based on the above results for $\mathcal{H}$-tensors, new sufficient conditions for testing the positive definiteness of even-order real symmetric tensors are provided.

In view of Proposition 2, the following result is easily obtained.

Theorem 8.

Let $\mathcal{A}=\left(a_{i_1\cdots i_m}\right)\in C^{[m,n]}$ be an even-order symmetric tensor with $a_{ii\cdots i}>0(\forall i\in N)$. Tensor $\mathcal{A}$ is positive definite if $\mathcal{A}$ satisfies one of the following conditions:

$\left(i\right)$

the conditions of Theorem 4;

$\left(ii\right)$

the conditions of Theorem 5;

$\left(iii\right)$

the conditions of Theorem 6.

Example 4.

Consider the following sixth-degree homogeneous polynomial

$$\begin{array}{ccc}\hfill f\left(x\right)& =& \mathcal{A}{x}^6\hfill \\ & =& 8x_1^6+17x_2^6+200x_3^6+62x_4^6+10x_5^6+20x_6^6-6x_1{x}_2^5-30x_1{x}_3^5\hfill \\ & & -6x_1{x}_4^5-6x_2{x}_3^5-6x_2{x}_4^5-24x_3{x}_4^5-20x_2^3{x}_3^3+6x_1^5{x}_4,\hfill \end{array}$$

where $x={(x_1,x_2,x_3,x_4,x_5,x_6)}^T$ and $\mathcal{A}=\left(a_{i_1{i}_2{i}_3{i}_4{i}_5{i}_6}\right)\in C^{[6,6]}$ is a real symmetric tensor with elements defined as follows:

$$\begin{array}{ccc}\hfill a_{111111}& =& 8,a_{222222}=17,a_{333333}=200,\hfill \\ \hfill a_{444444}& =& 62,a_{555555}=10,a_{666666}=20,\hfill \\ \hfill a_{122222}& =& a_{212222}=a_{221222}=a_{222122}=a_{222212}=a_{222221}=-1,\hfill \\ \hfill a_{133333}& =& a_{313333}=a_{331333}=a_{333133}=a_{333313}=a_{333331}=-5,\hfill \\ \hfill a_{144444}& =& a_{414444}=a_{441444}=a_{444144}=a_{444414}=a_{444441}=-1,\hfill \\ \hfill a_{233333}& =& a_{323333}=a_{332333}=a_{333233}=a_{333323}=a_{3333332}=-1,\hfill \\ \hfill a_{244444}& =& a_{424444}=a_{442444}=a_{444244}=a_{444424}=a_{444442}=-1,\hfill \\ \hfill a_{344444}& =& a_{434444}=a_{444344}=a_{444344}=a_{444434}=a_{444443}=-4,\hfill \\ \hfill a_{222333}& =& a_{223233}=a_{223323}=a_{223332}=a_{232233}=a_{232323}=a_{232332}=-1,\hfill \\ \hfill a_{233223}& =& a_{233232}=a_{233322}=a_{333222}=a_{332322}=a_{332232}=a_{332223}=-1,\hfill \\ \hfill a_{323322}& =& a_{323232}=a_{323223}=a_{322332}=a_{322323}=a_{322233}=-1,\hfill \\ \hfill a_{411111}& =& a_{141111}=a_{114111}=a_{111411}=a_{111141}=a_{111141}=1,\hfill \end{array}$$

and other $a_{i_1{i}_2{i}_3{i}_4{i}_5{i}_6}=0$. Since

$$\begin{array}{ccc}\hfill R_1\left(\mathcal{A}\right)& =& 12,R_2\left(\mathcal{A}\right)=17,R_3\left(\mathcal{A}\right)=44,\hfill \\ \hfill R_4\left(\mathcal{A}\right)& =& 31,R_5\left(\mathcal{A}\right)=0,R_6\left(\mathcal{A}\right)=0,\hfill \end{array}$$

then $N_3\left(\mathcal{A}\right)=\{3,4,5,6\}$, $N_2\left(\mathcal{A}\right)=\left\{1\right\}$, $N_1\left(\mathcal{A}\right)=\left\{2\right\}$. By calculations, we have

$$\begin{array}{ccc}\hfill y_1& =& \frac{2}{3},m=\frac{2}{3},{\omega }_0=\frac{1}{2},{\omega }_1=\frac{21}{62},h_2=\frac{961}{1449},\hfill \\ \hfill {\theta }_{2,3}& =& \frac{641}{3100},{\theta }_{2,4}=\frac{551}{1922},P_{2,3}=\frac{1282}{31},P_{2,4}=\frac{551}{31},\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \sum _{\underset{{\delta }_{i{i}_2\cdots i_m}=0}{i_2{i}_3{i}_4{i}_5{i}_6\in N_2^5}}\underset{j\in \left\{i_2,i_3,i_4,i_5,i_6\right\}}{max}\left\{y_j\right\}\left|a_{1i_2{i}_3{i}_4{i}_5{i}_6}\right|+\delta \sum _{i_2{i}_3{i}_4{i}_5{i}_6\in N_0^5}\left|a_{1i_2{i}_3{i}_4{i}_5{i}_6}\right|\hfill \\ & & +h_{k+1}\sum _{i_2{i}_3{i}_4{i}_5{i}_6\in N_3^5}\underset{j\in \left\{i_2,i_3,i_4,i_5,i_6\right\}}{max}\left\{{\theta }_{k+1,j}\right\}\left|a_{1i_2{i}_3{i}_4{i}_5{i}_6}\right|\hfill \\ & =& 6\times \frac{2}{3}+6\times \frac{961}{1449}\times \frac{551}{1922}=\frac{\mathit{2},\mathit{386},\mathit{163}}{\mathit{464},\mathit{163}}<8\times \frac{2}{3}=\frac{16}{3}=\left|a_{111111}\right|y_1.\hfill \end{array}$$

So, $\mathcal{A}$ satisfies the conditions of Theorem 4. By Theorem 8, we have that $\mathcal{A}$ is positive definite; that is, $f\left(x\right)$ is positive definite.

However, for $i=1$, we have

$$\begin{array}{c}\hfill a_{111111}=8<12=R_1\left(\mathcal{A}\right),\end{array}$$

$$\begin{array}{c}\hfill a_{444444}\left(a_{111111}-R_1\left(\mathcal{A}\right)+|a_{144444}|\right)=-186<31=R_4\left(\mathcal{A}\right)|a_{144444}|,\end{array}$$

$$\begin{array}{c}\hfill \left|a_{111111}\right|=8<9=\sum _{\underset{{\delta }_{2i_2{i}_3{i}_4{i}_5{i}_6}=0}{i_2{i}_3{i}_4{i}_5{i}_6\in N^5/N_3^5}}\left|a_{1i_2{i}_3{i}_4{i}_5{i}_6}\right|+\sum _{i_2{i}_3{i}_4{i}_5{i}_6\in N_3^5}\underset{j\in \{i_2,i_3,i_4,i_5,i_6\}}{max}\frac{R_j\left(\mathcal{A}\right)}{\left|a_{jjjjjj}\right|}\left|a_{1i_2{i}_3{i}_4{i}_5{i}_6}\right|,\end{array}$$

and

$$\begin{array}{ccc}\hfill |a_{111111}|r_1=\frac{8}{3}<\frac{4867}{651}& =& \sum _{i_2{i}_3{i}_4{i}_5{i}_6\in N_0^5}|a_{1i_2{i}_3{i}_4{i}_5{i}_6}|+\sum _{\genfrac{}{}{0pt}{}{i_2{i}_3{i}_4{i}_5{i}_6\in N_2^5}{{\delta }_{1i_2{i}_3{i}_4{i}_5{i}_6}=0}}\underset{j\in \{i_2,i_3,i_4,i_5,i_6\}}{max}\left\{r_j\right\}|a_{1i_2{i}_3{i}_4{i}_5{i}_6}|\hfill \\ & & +\sum _{i_2{i}_3{i}_4{i}_5{i}_6\in N_3^5}\underset{j\in \{i_2,i_3,i_4,i_5,i_6\}}{max}\left\{t_j\right\}|a_{1i_2{i}_3{i}_4{i}_5{i}_6}|.\hfill \end{array}$$

Hence, we cannot use Theorem 3 in in [19], Theorem 4 in in [20], Theorem 1 in [17], and Theorem 1 in [18] to identify the positive definiteness of $\mathcal{A}$, respectively.

6. Conclusions

In this paper, we presented new criterions for $\mathcal{H}$-tensors, which are used for testing the positive definiteness of an homogeneous polynomial. We also proposed a new non-parameter iterative algorithm for $\mathcal{H}$-tensors, which can stop within finite steps. These methods were expressed in terms of the elements of $\mathcal{A}$, so they can be checked easily.