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BY 4.0 license Open Access Published by De Gruyter Open Access June 25, 2022

An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms

  • Dongjian Bai and Feng Wang EMAIL logo
From the journal Open Mathematics
https://doi.org/10.1515/math-2022-0043
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Abstract

A positive definite homogeneous multivariate form plays an important role in the field of optimization, and positive definiteness of the form can be identified by a special structured tensor. In this paper, based on the equivalence between the form and the corresponding tensor, and the links of the positive definiteness of a tensor with ℋ-tensor, we propose an ℋ-tensor-based criterion for identifying the positive definiteness of multivariate homogeneous forms. Some numerical examples are provided to illustrate the efficiency and validity of our results.

Keywords: homogeneous multivariate form; positive definiteness; ℋ-tensors; diagonally dominant tensors; irreducible; non-zero element chain
MSC 2010: 15A69; 15A18; 65F15; 65H17

1 Introduction

Let R ( C ) be the real(complex) field, N = { 1 , 2 , … , n } . An m th-order n -dimensional real(complex) tensor A = ( a j 1 j 2 ⋯ j m ) consists of n m real(complex) entrise:

a j 1 j 2 ⋯ j m ∈ R ( C ) ,

where j i = 1 , 2 , … , n , i = 1 , 2 , … , m [1,2,3, 4,5,6]. Obviously, a matrix is a 2nd-order tensor. Moreover, tensor A = ( a j 1 j 2 ⋯ j m ) is called symmetric [7] if

a j 1 j 2 ⋯ j m = a π ( j 1 j 2 ⋯ j m ) , ∀ π ∈ Π m ,

where Π m is the permutation group of m indices. If a j 1 j 2 ⋯ j m ≥ 0 , then tensor A is called a nonnegative tensor. Tensor ℐ = ( δ j 1 j 2 ⋯ j m ) is called the unit tensor [8], where

δ j 1 j 2 ⋯ j m = 1 , if j 1 = ⋯ = j m , 0 , otherwise .

Denote an m -th degree multivariate homogeneous form of n variables f ( x ) as follows:

(1) f ( x ) = ∑ j 1 , j 2 , … , j m ∈ N a j 1 j 2 ⋯ j m x j 1 x j 2 ⋯ x j m ,

where x ∈ R n . While m is even, f ( t ) is labeled positive definite if

f ( t ) > 0 , for any t ∈ R n , t ≠ 0 .

Function f ( x ) in (1) can be expressed as the tensor product of an m th-order n -dimensional symmetric tensor A and x m defined as follows:

(2) f ( x ) ≡ A x m = ∑ j 1 , j 2 , … , j m ∈ N a j 1 j 2 ⋯ j m x j 1 x j 2 ⋯ x j m ,

where x ∈ R n and x m is an m th-order n -dimensional rank-one tensor with entries x j 1 ⋯ x j m [5]. The symmetric tensor A is positive definite if f ( x ) is positive definite [9].

The positive definiteness of tensor has received much attention of researchers’ in recent decade [10,11,12]. By the Sturm theorem, the positive definiteness of a multivariate homogeneous form can be identified for n ≤ 3 [13]. Ni et al. [9] presented an eigenvalue method for checking positive definiteness of a symmetric tensor. While, all the eigenvalues should be calculated in this method, and it is not practical when tensor order or dimension is large.

Recently, Li et al. [14] provided an ℋ -tensor-based method for identifying the positive definiteness of an even-order symmetric tensor. It is well known that an even-order symmetric ℋ -tensor with positive diagonal entries is positive definite. Hence, we can check the positive definiteness of a tensor via the aid of ℋ -tensor. Subsequently, with the help of generalized diagonal dominance, miscellaneous criterions for ℋ -tensors and ℳ -tensors are provided [15,16,17, 18,19,20], which depends on the entries of tensors and is efficient to determine ℋ -tensor ( ℳ -tensor).

Theorem 1

[20] Let A = ( a j 1 ⋯ j m ) ∈ C [ m , n ] . If

∣ a j j ⋯ j ∣ r j > ∑ j 2 j 3 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 j 3 ⋯ j m ∈ N 2 m − 1 δ j j 2 ⋯ j m = 0 max t ∈ { j 2 , j 3 , … , j m } { r t } ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 max t ∈ { j 2 , j 3 , … , j m } { l t } ∣ a j j 2 ⋯ j m ∣ , ∀ j ∈ N 2

and

∣ a i i ⋯ i ∣ ≠ ∑ i 2 i 3 ⋯ i m ∈ N 0 m − 1 δ i i 2 ⋯ i m = 0 ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N 1 ≠ ∅ ( o r N 1 = ∅ ) ,

then A is an ℋ -tensor.

In this paper, we continue to present new criteria for ℋ -tensors. These new results improve the corresponding conclusions [20,21,22]. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. Several numerical experiments demonstrate its efficiency.

Several specially definitions extended from matrices are as follows.

Definition 1

[17] Let A = ( a j 1 j 2 ⋯ j m ) be a complex tensor of order m dimension n . A is an ℋ -tensor if there exists a positive vector x = ( x 1 , x 2 , … , x n ) T ∈ R n , such that

∣ a j j ⋯ j ∣ x j m − 1 > ∑ j 2 , … , j m ∈ N δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ x j 2 ⋯ x j m , ∀ j ∈ N .

Definition 2

[8] A complex tensor A = ( a j 1 j 2 ⋯ j m ) of order m dimension n is called reducible if there is a nonempty subset I ⊂ N , such that

a j 1 j 2 ⋯ j m = 0 , ∀ j 1 ∈ I , ∀ j 2 , … , j m ∉ I .

Otherwise, A is called irreducible.

Definition 3

[18] Let A = ( a j 1 j 2 ⋯ j m ) be a complex tensor of order m dimension n . For i , j ∈ N ( i ≠ j ) , if there are indices k 1 , k 2 , … , k r , such that

∑ j 2 , … , j m ∈ N δ k s j 2 … j m = 0 , k s + 1 ∈ { j 2 , … , j m } ∣ a k s j 2 ⋯ j m ∣ ≠ 0 , s = 0 , 1 , … , r .

where k 0 = j , k r + 1 = i , and then there exists a nonzero elements chain from j to i .

The remainder of this paper is as follows. In Section 2, we gives some criteria for the identification of ℋ -tensors. In Section 3, some new conditions for the identification of positive definite tensors are presented. Some numerical experiments are also provided to show the effectiveness of new methods.

2 Criteria for ℋ -tensors

In this section, three new criteria for identifying ℋ -tensors are presented. First, for the convenience of description, some notation and lemmas are given. For a tensor A = ( a j 1 j 2 ⋯ j m ) of order m dimension n , denote

S m − 1 = { j 2 j 3 ⋯ j m : j i ∈ S , i = 2 , 3 , ⋯ , m } , ∅ ≠ S ⊂ N ; N m − 1 ⧹ S m − 1 = { j 2 j 3 ⋯ j m : j 2 j 3 ⋯ j m ∈ N m − 1 and j 2 j 3 ⋯ j m ∉ S m − 1 } ; N 0 m − 1 = N m − 1 ⧹ ( N 2 m − 1 ∪ N 3 m − 1 ) ; Λ j ( A ) = ∑ j 2 , … , j m ∈ N δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ = ∑ j 2 , … , j m ∈ N ∣ a j j 2 ⋯ j m ∣ − ∣ a j j ⋯ j ∣ ; N 1 = { j ∈ N : 0 < ∣ a j j ⋯ j ∣ = Λ j ( A ) } ; N 2 = { j ∈ N : 0 < ∣ a j j ⋯ j ∣ < Λ j ( A ) } ; N 3 = { j ∈ N : ∣ a j j ⋯ j ∣ > Λ j ( A ) } ; w j = Λ j ( A ) Λ j ( A ) + ∣ a j j ⋯ j ∣ , w = max { w j } , ∀ j ∈ N 2 ; r = max j ∈ N 3 ∑ j 2 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ ∣ a j j ⋯ j ∣ − ∑ j 2 … j m ∈ N 3 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ ;

P j , r ( A ) = ∑ j 2 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ + r ∑ j 2 ⋯ j m ∈ N 3 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ , j ∈ N 3 ; r 1 = max j ∈ N 3 ∑ j 2 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ ∣ a j j ⋯ j ∣ − ∑ j 2 … j m ∈ N 3 m − 1 δ j j 2 … i m = 0 max t ∈ { j 2 , j 3 , … , j m } P t , r ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ ; μ = max { w j , r 1 } , j ∈ N 2 ;

P j , r 1 ( A ) = ∑ j 2 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ + r 1 ∑ j 2 ⋯ j m ∈ N 3 m − 1 δ j j 2 … j m = 0 max t ∈ { j 2 , j 3 , … , j m } P t , r ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ , j ∈ N 3 ; h = max j ∈ N 3 μ ∑ j 2 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ w P j , r 1 ( A ) − ∑ j 2 … j m ∈ N 3 m − 1 δ j j 2 … j m = 0 max t ∈ { j 2 , j 3 , … , j m } P t , r 1 ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ .

Throughout this paper, we assume that N 1 ≠ ∅ and N 2 ≠ ∅ . Meanwhile, we also assume that tensor A satisfies: a j j ⋯ j ≠ 0 , Λ j ( A ) ≠ 0 , ∀ j ∈ N .

Lemma 1

[17] If tensor A = ( a j 1 j 2 ⋯ j m ) is strictly diagonally dominant, then A is an ℋ -tensor.

Lemma 2

[14] Let A = ( a j 1 j 2 ⋯ j m ) be an mth-order n-dimensional complex tensor. If there is a positive diagonal matrix X such that A X m − 1 is an ℋ -tensor, then A is an ℋ -tensor.

Lemma 3

[14] Let A = ( a j 1 j 2 ⋯ j m ) be an mth-order n-dimensional complex tensor. If A is irreducible,

∣ a j j ⋯ j ∣ ≥ Λ j ( A ) , ∀ j ∈ N ,

and N 3 ( A ) ≠ ∅ , then A is an ℋ -tensor.

Lemma 4

[18] Let A = ( ( a j 1 j 2 ⋯ j m ) ) be an mth-order n-dimensional complex tensor. If

  1. ∣ a j j ⋯ j ∣ ≥ Λ j ( A ) , ∀ j ∈ N ;

  2. N 3 = { j ∈ N : ∣ a j j ⋯ j ∣ > Λ j ( A ) } ≠ ∅ ;

  3. For ∀ j ∉ N 1 , there exists a nonzero elements chain from j to i such that i ∈ N 1 ,

then A is an ℋ -tensor.

Next, we give some new criteria for ℋ -tensors.

Theorem 2

Let A = ( a j 1 j 2 ⋯ j m ) be an mth-order n-dimensional complex tensor. If A satisfies

(3) ∣ a j j ⋯ j ∣ > μ w j ∑ j 2 j 3 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 j 3 ⋯ j m ∈ N 2 m − 1 δ j j 2 ⋯ j m = 0 ∣ a j j 2 ⋯ j m ∣ w w j + h w j ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 max t ∈ { j 2 , j 3 , … , j m } P t , r 1 ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ , ∀ j ∈ N 2 ,

and there exists j 2 j 3 ⋯ j m ∈ N m − 1 ⧹ N 1 m − 1 for any j ∈ N 1 such that a j j 2 j 3 ⋯ j m ≠ 0 , then A is an ℋ -tensor.

Proof

By 0 ≤ r 1 ≤ r < 1 , according to the definition of r , P i , r ( A ) , r 1 and P i , r 1 ( A ) , for any i ∈ N 3 ,

r 1 ∣ a i i ⋯ i ∣ ≥ ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ + r 1 ∑ i 2 … i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ ,

that is, P i , r 1 ( A ) ≤ r 1 ∣ a i i ⋯ i ∣ ( ∀ i ∈ N 3 ) , so

0 < P i , r 1 ( A ) ∣ a i i ⋯ i ∣ ≤ r 1 ≤ r < 1 , ∀ i ∈ N 3 .

By the definitions of w i and μ , for any i ∈ N 3 ,

μ ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ w P i , r 1 ( A ) − ∑ i 2 … i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ ≤ μ P i , r 1 ( A ) − r 1 ∑ i 2 ⋯ i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ P i , r 1 ( A ) − ∑ i 2 … i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ < 1 .

For ∀ i ∈ N 3 , from the definition of h , we have 0 < h < 1 , and

(4) h P i , r 1 ( A ) ≥ μ ∑ i 2 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ w + h ∑ i 2 … i m ∈ N 3 m − 1 δ i i 2 … i m = 0 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ .

By inequality (3), for all i ∈ N 2 , we have

w i ∣ a i i ⋯ i ∣ > μ ∑ i 2 i 3 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 i 3 ⋯ i m ∈ N 2 m − 1 δ i i 2 ⋯ i m = 0 ∣ a i i 2 ⋯ i m ∣ w + h ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ .

Let

(5) R j ≡ 1 ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 ∣ a j j 2 ⋯ j m ∣ w j ∣ a j j ⋯ j ∣ − μ ∑ j 2 j 3 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 j 3 ⋯ j m ∈ N 2 m − 1 δ j j 2 ⋯ j m = 0 ∣ a j j 2 ⋯ j m ∣ w + h ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 max t ∈ { j 2 , j 3 , … , j m } P t , r 1 ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ , ∀ j ∈ N .

If ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 ∣ a j j 2 ⋯ j m ∣ = 0 , we denote R j = + ∞ . By Equality (5), we have

R j > 0 ( ∀ j ∈ N 2 ) , 0 < h P t , r 1 ( A ) ∣ a t t ⋯ t ∣ < 1 ( ∀ t ∈ N 3 ) .

Hence, there exists a positive number ε such that

0 < ε < min j ∈ N 2 R j ≤ + ∞ , max t ∈ N 3 h P t , r 1 ( A ) ∣ a t t ⋯ t ∣ + ε < 1 .

Let the matrix D = diag ( d 1 , d 2 , … , d n ) , denote ℬ = A D m − 1 = ( b i 1 i 2 ⋯ i m ) , where

d i = ( μ ) 1 m − 1 , i ∈ N 1 , ( w i ) 1 m − 1 , i ∈ N 2 , ε + h P i , r 1 ( A ) ∣ a i i ⋯ i ∣ 1 m − 1 , i ∈ N 3 .

For ∀ j ∈ N 1 , there exists j 2 j 3 ⋯ j m ∈ N m − 1 ⧹ N 1 m − 1 such that a j j 2 j 3 ⋯ j m ≠ 0 , and for ∀ t ∈ N 3 , let ε > 0 satisfy 0 < ε + h P t , r 1 ( A ) ∣ a t t ⋯ t ∣ < μ < 1 , we obtain

Λ j ( ℬ ) = μ ∑ j 2 ⋯ j m ∈ N 0 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ w + ∑ j 2 … j m ∈ N 3 m − 1 ∣ a j j 2 ⋯ j m ∣ ε + h P j 2 , r 1 ( A ) ∣ a j 2 j 2 ⋯ j 2 ∣ 1 m − 1 ⋯ ε + h P j m , r 1 ( A ) ∣ a j m j m ⋯ j m ∣ 1 m − 1 ≤ μ ∑ j 2 ⋯ j m ∈ N 0 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ w + ∑ j 2 … j m ∈ N 3 m − 1 ∣ a j j 2 ⋯ j m ∣ ε + max t ∈ { j 2 , j 3 , … , j m } h P t , r 1 ( A ) ∣ a t t ⋯ t ∣ < μ ∑ j 2 ⋯ j m ∈ N 0 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 … j m ∈ N 3 m − 1 ∣ a j j 2 ⋯ j m ∣ = μ Λ j ( A ) = μ ∣ a j j … j ∣ = ∣ b j j … j ∣ .

For ∀ j ∈ N 2 , by equality (5), we have

Λ j ( ℬ ) = μ ∑ j 2 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ w + ∑ j 2 … j m ∈ N 3 m − 1 ∣ a j j 2 ⋯ j m ∣ ε + h P j 2 , r 1 ( A ) ∣ a j 2 j 2 ⋯ j 2 ∣ 1 m − 1 ⋯ ε + h P j m , r 1 ( A ) ∣ a j m j m ⋯ j m ∣ 1 m − 1 ≤ μ ∑ j 2 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ w + ∑ j 2 … j m ∈ N 3 m − 1 ∣ a j j 2 ⋯ j m ∣ ε + max t ∈ { j 2 , j 3 , … , j m } h P t , r 1 ( A ) ∣ a t t ⋯ t ∣ = ε ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 ∣ a j j 2 ⋯ j m ∣ + μ ∑ j 2 j 3 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 j 3 ⋯ j m ∈ N 2 m − 1 δ j j 2 ⋯ j m = 0 ∣ a j j 2 ⋯ j m ∣ w + h ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 max t ∈ { j 2 , j 3 , … , j m } P t , r 1 ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ < R j ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 ∣ a j j 2 ⋯ j m ∣ + μ ∑ j 2 j 3 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 j 3 ⋯ j m ∈ N 2 m − 1 δ j j 2 ⋯ j m = 0 ∣ a j j 2 ⋯ j m ∣ w + h ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 max t ∈ { j 2 , j 3 , … , j m } P t , r 1 ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ = w j ∣ a j j … j ∣ = ∣ b j j … j ∣ .

Finally, for j ∈ N 3 , by inequality (4), we have

Λ j ( ℬ ) = μ ∑ j 2 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ w + ∑ j 2 … j m ∈ N 3 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ ε + h P j 2 , r 1 ( A ) ∣ a j 2 j 2 ⋯ j 2 ∣ 1 m − 1 ⋯ ε + h P j m , r 1 ( A ) ∣ a j m j m ⋯ j m ∣ 1 m − 1 ≤ μ ∑ j 2 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ w + ∑ j 2 … j m ∈ N 3 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ ε + max t ∈ { j 2 , j 3 , … , j m } h P t , r 1 ( A ) ∣ a t t ⋯ t ∣ = ε ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ + μ ∑ i 2 i 3 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 j 3 ⋯ j m ∈ N 2 m − 1 ∣ a j j 2 ⋯ j m ∣ w + h ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 δ j j 2 ⋯ j m = 0 max t ∈ { j 2 , j 3 , … , j m } P t , r 1 ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ ≤ ε ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 δ j j 2 … j m = 0 ∣ a j j 2 ⋯ j m ∣ + h P j , r 1 ( A ) < ε ∣ a j j … j ∣ + h P j , r 1 ( A ) = ∣ b j j … j ∣ .

Therefore, ∣ b j j ⋯ j ∣ > Λ j ( ℬ ) ( ∀ i ∈ N ) . So ℬ is an ℋ -tensor by Lemma 1. From Lemma 2, A is an ℋ -tensor.□

Remark 1

It is hard to theoretically give the comparison between our result and the known ones in [20]. The following numerical example illustrates that our proposed criteria is more effective to theirs in some cases. Moreover, we provide an example that satisfies our conditions in Theorem 2 but not those in Theorem 1 of [20].

Let

K ( A ) = j ∈ N 2 : ∣ a j j ⋯ j ∣ > μ w j ∑ j 2 j 3 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 j 3 ⋯ j m ∈ N 2 m − 1 δ j j 2 ⋯ j m = 0 ∣ a j j 2 ⋯ j m ∣ w w j + h w j ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 max t ∈ { j 2 , j 3 , … , j m } P t , r 1 ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ .

Theorem 3

Let A = ( a j 1 j 2 ⋯ j m ) be an mth-order n-dimensional complex tensor. If A is irreducible,

(6) ∣ a j j ⋯ j ∣ ≥ μ w j ∑ j 2 j 3 ⋯ j m ∈ N 0 m − 1 ∣ a j j 2 ⋯ j m ∣ + ∑ j 2 j 3 ⋯ j m ∈ N 2 m − 1 δ j j 2 ⋯ j m = 0 ∣ a j j 2 ⋯ j m ∣ w w j + h w j ∑ j 2 j 3 ⋯ j m ∈ N 3 m − 1 max t ∈ { j 2 , j 3 , … , j m } P t , r 1 ( A ) ∣ a t t ⋯ t ∣ ∣ a j j 2 ⋯ j m ∣ , ∀ j ∈ N 2 ,

and K ( A ) ≠ ∅ , then A is an ℋ -tensor.

Proof

Since A is irreducible, so

∑ i 2 i 3 ⋯ i m ∈ N m − 1 ⧹ N 3 m − 1 ∣ a i i 2 ⋯ i m ∣ > 0 , ∀ i ∈ N 3 .

By inequality (6), we obtain

(7) ∣ a i i ⋯ i ∣ w i ≥ μ ∑ i 2 i 3 ⋯ i m ∈ N 0 m − 1 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 i 3 ⋯ i m ∈ N 2 m − 1 δ i i 2 ⋯ i m = 0 ∣ a i i 2 ⋯ i m ∣ w + h ∑ i 2 i 3 ⋯ i m ∈ N 3 m − 1 max j ∈ { i 2 , i 3 , … , i m } P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ , ∀ i ∈ N 2 .

and at least one strict inequality in (7) holds.

Let the matrix D = diag ( d 1 , d 2 , … , d n ) , denote ℬ = A D m − 1 = ( b i 1 i 2 ⋯ i m ) , where

d i = ( μ ) 1 m − 1 , i ∈ N 1 , ( w i ) 1 m − 1 , i ∈ N 2 , h P i , r 1 ( A ) ∣ a i i ⋯ i ∣ 1 m − 1 , i ∈ N 3 .

For ∀ i ∈ N 1 , by μ > h P i , r 1 ( A ) ∣ a i i ⋯ i ∣ ( ∀ i ∈ N 3 ) , we have

Λ i ( ℬ ) = μ ∑ i 2 ⋯ i m ∈ N 0 m − 1 δ i i 2 … i m = 0 ∣ a i i 2 ⋯ i m ∣ + ∑ i 2 ⋯ i m ∈ N 2 m − 1 ∣ a i i 2 ⋯ i m ∣ w + ∑ i 2 … i m ∈ N 3 m − 1 ∣ a i i 2 ⋯ i m ∣ h P i 2 , r 1 ( A ) ∣ a i 2 i 2 ⋯ i 2 ∣ 1 m − 1 ⋯ h P i m , r 1 ( A ) ∣ a i m i m ⋯ i m ∣ 1 m − 1 ≤ μ ∑ i 2