New iterative criteria for strong H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-tensors and an application

Strong H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-tensors play an important role in identifying the positive definiteness of even-order real symmetric tensors. In this paper, some new iterative criteria for identifying strong H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-tensors are obtained. These criteria only depend on the elements of the tensors, and it can be more effective to determine whether a given tensor is a strong H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}$\end{document}-tensor or not by increasing the number of iterations. Some numerical results show the feasibility and effectiveness of the algorithm.


Introduction
A tensor can be regarded as a higher-order generalization of a matrix. Let C(R) denote the set of all complex (real) numbers and N = {, , . . . , n}. We call A = (a i  i  ···i m ) an mth-order n-dimensional complex (real) tensor, if where i j = , , . . . , n for j = , , . . . , m [, ]. Obviously, a vector is a tensor of order  and a matrix is a tensor of order . A tensor A = (a i  i  ···i m ) is called symmetric [], if where m is the permutation group of m indices. Furthermore, an mth-order n-dimensional tensor I = (δ i  i  ···i m ) is called the unit tensor [], if its entries Let A = (a i  i  ···i m ) be an mth-order n-dimensional complex tensor. If there exist a number λ ∈ C and a non-zero vector x = (x  , x  , . . . , x n ) T ∈ C n that are solutions of the following homogeneous polynomial equations: if(i  , i  , . . . , i m ) = (i  , i  , . . . , i  ); We call A irreducible if A is not reducible.
be an mth-order n-dimensional tensor and a n-by-n matrix X = (x ij ) on mode-k is defined According to Definition ., we denote Particularly, for X = diag(x  , x  , . . . , x n ), the product of the tensor A and the matrix X is given by where k  = i, k r+ = j, we call there is a non-zero elements chain from i to j.
For an mth degree homogeneous polynomial of n variables f (x) is denoted as The homogeneous polynomial f (x) in (.) is equivalent to the tensor product of an mthorder n-dimensional symmetric tensor A and x m defined by []

Lemma . ([]) Let A be an even-order real symmetric tensor, then A is positive definite if and only if all of its H-eigenvalues are positive.
Although from Lemma . we can verify the positive definiteness of an even-order symmetric tensor A (the positive definiteness of the mth-degree homogeneous polynomial f (x)) by computing the H-eigenvalues of A. In [-], for a non-negative tensor, some algorithms are provided to compute its largest eigenvalue. And in [, ], based on semidefinite programming approximation schemes, some algorithms are also given to compute eigenvalues for general tensors with moderate sizes. However, it is difficult to compute all these H-eigenvalues when m and n are large. Recently, by introducing the definition of strong H-tensor [, ], Li et al.
[] provided a practical sufficient condition for identifying the positive definiteness of an even-order symmetric tensor (see Lemma .).
be an even-order real symmetric tensor with a k···k >  for all k ∈ N . If A is a strong H-tensor, then A is positive definite.

As mentioned in []
, it is still difficult to determine a strong H-tensor in practice by using the definition of strong H-tensor because the conditions 'there is a positive vector x = (x  , x  , . . . , x n ) T ∈ R n such that, for all i ∈ N , the Inequality (.) holds' in Definition . is unverifiable for there are an infinite number of positive vector in R n . Therefore, much literature has focused on researching how to determine that a given tensor is a strong Htensor by using the elements of the tensors without Definition . recently, consequently, the corresponding even-order real symmetric tensor is positive definite. Therefore, the main aim of this paper is to study some new iterative criteria for identifying strong Htensors only depending on the elements of the tensors.
Before presenting our results, we review the existing ones that relate to the criteria for strong H-tensors. Let S be an arbitrary nonempty subset of N and let N\S be the complement of S in N . Given an mth-order n-dimensional complex tensor A = (a i  i  ···i m ), we denote In [], Wang and Sun derived the following result.
then A is a strong H-tensor.
In the sequel, Wang et al. in [] proved the following result.

complex tensor with order m and dimension n. If for all
then A is a strong H-tensor. Now, some notations are given, which will be used throughout this paper. Let The remainder of the paper is organized as follows. In Section ., some criteria for identifying strong H-tensors are obtained; as an interesting application of these criteria, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are presented in Section .. Numerical examples are given to verify the corresponding results. Finally, some conclusions are given to end this paper in Section .
We adopt the following notations throughout this paper. The calligraphy letters A, B, H, . . . denote tensors; the capital letters A, B, D, . . . represent matrices; the lowercase letters x, y, . . . refer to vectors.

Criteria for identifying strong H-tensors
In this subsection, we give some new criteria for identifying strong H-tensors by making use of elements of tensors only. For the convenience of our discussion, we start with the following lemmas, which will be useful in the next proofs.
Lemma . Let A = (a i  i  ···i m ) be an mth-order n-dimensional complex tensor, then, for all i ∈ N  , l = , , . . . , From the above inequality, ∀i ∈ N  , we obtain Together with the expression of R () i (A), for ∀i ∈ N  , we deduce that Combining the expression of h () and the above inequality results in Besides, for ∀i ∈ N  , Dividing by |a ii···i | on both sides of the above inequality yields Furthermore, by the expression of R () i (A) and the above inequality, for i ∈ N  , we have Combining the expression of h () and the above inequality results in In the same manner as applied in the proof of (.), for i ∈ N  , we obtain For i ∈ N  , it follows from inequalities (.) and (.) that By an analogical proof as above, we can derive that, for i ∈ N  , l = , , . . . , The proof is completed. A is a strictly diagonally dominant tensor, then A is a strong H-tensor.

Lemma . ([]) If
By Lemma ., if N  = ∅ (A is a strictly diagonally dominant tensor), then A is a strong H-tensor; by Lemma ., if A is a strong H-tensor, then N  = ∅. Hence, we always assume that N  = ∅, N  = ∅. In addition, we also assume that A satisfies a ii···i = , r i (A) = , ∀i ∈ N .
and strictly inequality holds for at least one i, then A is a strong H-tensor.

Lemma . ([]) Let
then A is a strong H-tensor.
Proof By the expression of h (l+) , it follows that Together with Inequality (.), there exists a ε > , sufficiently small such that for all i ∈ N  , and for all i ∈ N  , Let the matrix X = diag(x  , x  , . . . , x n ), where We see by Inequality (.) that (h (l+) δ (l+) i + ε)  m- <  (∀i ∈ N  ), as ε = ∞, x i = ∞, which shows that X is a diagonal matrix with positive entries. Let B = AX m- . Next, we will prove that B is strictly diagonally dominant.
For any i ∈ N  , it follows from (.) that For any i ∈ N  , it follows from (.) that Therefore, from the above inequalities, we conclude that |b ii···i | > r i (B) for all i ∈ N , B is strictly diagonally dominant, and by Lemma ., B is a strong H-tensor. Further, by Lemma ., A is a strong H-tensor.
in addition, the strict inequality holds for at least one i ∈ N  , then A is a strong H-tensor.
Proof Notice that A is irreducible; this implies Let the matrix X = diag(x  , x  , . . . , x n ), where Adopting the same procedure as in the proof of Theorem ., we conclude that |b i···i | ≥ r i (B) for all i ∈ N . Moreover, the strict inequality holds for at least one i ∈ N  , thus, there exists at least an i ∈ N such that |b ii···i | > r i (B).
On the other hand, since A is irreducible, and so is B. Then by Lemma ., we see that B is a strong H-tensor. By Lemma ., A is also a strong H-tensor.
and if ∀i ∈ N\J = ∅, there exists a non-zero elements chain from i to j such that j ∈ J = ∅, then A is a strong H-tensor.
Proof Let the matrix X = diag(x  , x  , . . . , x n ), where Similar to the proof of Theorem ., we can obtain |b ii···i | ≥ r i (B) for all i ∈ N , and there exists at least an i ∈ N  such that |b ii···i | > r i (B).
On the other hand, if |b ii···i | = r i (B), then i ∈ N\J; by the assumption, we know that there exists a non-zero elements chain of A from i to j, such that j ∈ J. Then there exists a nonzero elements chain of B from i to j, such that j satisfies |b jj···j | > r j (B).
Based on the above analysis, we conclude that the tensor B satisfies the conditions of Lemma ., so B is a strong H-tensor. By Lemma ., A is a strong H-tensor. The proof is completed. Remark . From Lemma ., we can also obtain smaller iterative coefficients h (l+) δ (l+) i by increasing l. Therefore, Theorem . in this paper can be more effective to determine whether a given tensor is a strong H-tensor or not by increasing the number of iterations.  iterations l = . When l = , we can get

Example . Consider a tensor
we see that A satisfies the conditions of Theorem ., then A is a strong H-tensor. In fact, there exists a positive diagonal matrix X = diag(, ., ., .) such that AX  is strictly diagonally dominant.

An application: the positive definiteness of an even-order real symmetric tensor
In this subsection, by making use of the results in Section ., we present new criteria for identifying the positive definiteness of an even-order real symmetric tensor. From Lemma . and Theorems .-., we easily obtain the following result.
be a th-degree homogeneous polynomial. We can get an th-order -dimensional real symmetric tensor A = (a i  i  i  i  ), where

Conclusions
In this paper, we give some criteria for identifying a strong H-tensor which only depend on the elements of tensors, and by increasing the number of iterations, we can determine whether a given tensor is a strong H-tensor or not more effective. We also present new criteria for identifying the positive definiteness of an even-order real symmetric tensor based on these criteria.