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

Dongjian Bai; Feng Wang

doi:10.1515/math-2022-0043

Article Open Access

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

Dongjian Bai and Feng Wang

Published/Copyright: June 25, 2022

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$Open Mathematics$

From the journal Open Mathematics Volume 20 Issue 1

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

∣ a j j ⋯ j ∣ ≥ Λ j ( A ) , ∀ j ∈ N ;
N 3 = { j ∈ N : ∣ a j j ⋯ j ∣ > Λ j ( A ) } ≠ ∅ ;
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 ⋯ 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 max j ∈ { i 2 , i 3 , … , i m } h P j , r 1 ( A ) ∣ a j j ⋯ j ∣ ∣ a i i 2 ⋯ i m ∣ < 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 ∣ + ∑ i 2 … i m ∈ N 3 m − 1 ∣ a i i 2 ⋯ i m ∣ = μ ∣ a i i … i ∣ = ∣ b i i … i ∣ .

For ∀ i ∈ N 2 , by inequality (7), we obtain

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

Next, we consider i ∈ N 3 , by inequality (4), we obtain

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

Therefore, ∣ b i i ⋯ i ∣ ≥ Λ i ( ℬ ) for all i ∈ N , and at least one strict inequality in (6) holds, that is, there exists an i 0 ∈ N 2 such that ∣ b i 0 i 0 ⋯ i 0 ∣ > Λ i 0 ( ℬ ) .

Notice that A is irreducible and so is ℬ . Hence, we obtain that ℬ is an ℋ -tensor by Lemma 3 and A is an ℋ -tensor by Lemma 2.□

Theorem 4

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

(8) ∣ 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 if any j ∈ N ⧹ K ( A ) ≠ ∅ , there is a nonzero elements chain from j to i such that i ∈ K ( A ) ≠ ∅ , then A is an ℋ -tensor.

Proof

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

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

By a similar proof to that of Theorem 3, we obtain that ∣ b j j ⋯ j ∣ ≥ Λ j ( ℬ ) ( ∀ j ∈ N ) , and there exists at least an i 0 ∈ N 2 such that ∣ b i 0 i 0 ⋯ i 0 ∣ > Λ i 0 ( ℬ ) .

On the other hand, if ∣ b j j ⋯ j ∣ = Λ j ( ℬ ) , then j ∈ N ⧹ K ( A ) , so there is a nonzero elements chain of A from j to i satisfying i ∈ K ( A ) . Hence, there is a nonzero elements chain of ℬ from j to i satisfying ∣ b i i ⋯ i ∣ > Λ i ( ℬ ) .

Therefore, we obtain that ℬ is an ℋ -tensor by Lemma 4 and A is an ℋ -tensor by Lemma 2.□

Below, we present a numerical example to show that the criteria for ℋ -tensor is obtained by only using the conditions of Theorem 2 but not the ones in [20].

Example 1

Given a 3rd-order three-dimensional tensor A = ( a i j k ) as follows:

A = [ A ( 1 , : , : ) , A ( 2 , : , : ) , A ( 3 , : , : ) ] , A ( 1 , : , : ) = 12 1 0 1 6 0 1 1 15 , A ( 2 , : , : ) = 1 1 0 0 4 0 0 0 6 , A ( 3 , : , : ) = 1 0 0 0 1 0 0 0 16 .

Obviously,

∣ a 111 ∣ = 12 , Λ 1 ( A ) = 25 , ∣ a 222 ∣ = 4 , Λ 2 ( A ) = 8 , ∣ a 333 ∣ = 16 , Λ 3 ( A ) = 2 ,

so N 1 = ∅ , N 2 = { 1 , 2 } , N 3 = { 3 } . By calculations, we have

w 1 = 25 25 + 12 = 25 37 , w 2 = 8 8 + 4 = 2 3 , w = 25 37 , r = 0 + 2 16 − 0 = 1 8 ,

P 3 , r ( A ) = 0 + 2 + 1 8 × 0 = 2 , r 1 = 0 + 2 16 − 1 8 × 0 = 1 8 , μ = 25 37 ,

P 3 , r 1 ( A ) = 0 + 2 + 1 8 × 1 8 × 0 = 2 , h = 25 37 × 0 + 25 37 × 2 2 − 1 8 × 0 = 25 37 .

When i = 1 , we obtain

μ ∑ i 2 i 3 ∈ N 0 2 ∣ a 1 i 2 i 3 ∣ + ∑ i 2 i 3 ∈ N 2 2 δ 1 i 2 i 3 = 0 ∣ a 1 i 2 i 3 ∣ w + h ∑ i 2 i 3 ∈ N 3 2 max j ∈ { i 2 , i 3 } P j , r 1 ( A ) ∣ a j j j ∣ ∣ a 1 i 2 i 3 ∣ = 25 37 × 2 + 25 37 × 8 + 25 37 × 1 8 × 15 = 2375 296 < 300 37 = 12 × 25 37 = ∣ a 111 ∣ × w 1 .

When i = 2 , we obtain

μ ∑ i 2 i 3 ∈ N 0 2 ∣ a 2 i 2 i 3 ∣ + ∑ i 2 i 3 ∈ N 2 2 δ 2 i 2 i 3 = 0 ∣ a 2 i 2 i 3 ∣ w + h ∑ i 2 i 3 ∈ N 3 2 max j ∈ { i 2 , i 3 } P j , r 1 ( A ) ∣ a j j j ∣ ∣ a 2 i 2 i 3 ∣ = 25 37 × 0 + 25 37 × 2 + 25 37 × 1 8 × 6 = 275 148 < 100 37 = 4 × 25 37 = ∣ a 111 ∣ × w 1 .

Hence, the conditions of Theorem 2 are satisfied, then A is an ℋ -tensor. However,

p 3 = 26 25 , s 3 = 13 200 , M 3 = 1 , F 3 = 26 25 , t 3 = 13 200 .

∑ i 2 i 3 ∈ N 0 2 ∣ a 1 i 2 i 3 ∣ + ∑ i 2 i 3 ∈ N 2 2 δ 1 i 2 i 3 = 0 max j ∈ { i 2 , i 3 } { r j } ∣ a 1 i 2 i 3 ∣ + ∑ i 2 i 3 ∈ N 3 2 max j ∈ { i 2 , i 3 } { t j } ∣ a 1 i 2 i 3 ∣ = 2 + 13 25 × 8 + 13 200 × 15 = 1427 200 156 25 = ∣ a 111 ∣ r 1 .

So, A does not satisfy the conditions of Theorem 1.

3 An application

In this section, we provide some ℋ -tensor-based criteria for identifying the positive definite tensor. First, we recall the following lemma.

Lemma 5

[14] Let A = ( a j 1 j 2 ⋯ j m ) be an mth-order n-dimensional even-order real symmetric tensor with a j j ⋯ j > 0 for ∀ j ∈ N . If A is an ℋ -tensor, then A is positive definite.

By Theorems 2–4 and Lemma 5, we can prove easily the following result.

Theorem 5

Let A = ( a j 1 j 2 ⋯ j m ) be an mth-order n-dimensional even-order real symmetric tensor a j j ⋯ j > 0 for ∀ j ∈ N . If one of the following conditions is satisfied:

All the conditions of Theorem 2;
All the conditions of Theorem 3;
All the conditions of Theorem 4,

then A is a positive definite tensor.

Example 2

Consider an 4th-degree homogeneous polynomial:

f ( x ) = A x 4 = 24 x 1 4 + 36 x 2 4 + 25 x 3 4 + 10 x 4 4 − 8 x 2 3 x 4 + 12 x 1 2 x 2 x 3 − 12 x 2 x 3 2 x 4 − 24 x 1 x 2 x 3 x 4 .

We can obtain an 4th-order four-dimensional real symmetric tensor A = ( a j 1 j 2 j 3 j 4 ) with entries

a 1111 = 24 , a 2222 = 36 , a 3333 = 25 , a 4444 = 10 , a 2224 = a 2242 = a 2422 = a 4222 = − 2 , a 1123 = a 1132 = a 1213 = a 1312 = a 1231 = a 1321 = 1 , a 2113 = a 2131 = a 2311 = a 3112 = a 3121 = a 3211 = 1 , a 2334 = a 2343 = a 2433 = a 4233 = a 4323 = a 4332 = − 1 , a 3234 = a 3243 = a 3324 = a 3342 = a 3423 = a 3432 = − 1 , a 1234 = a 1243 = a 1324 = a 1342 = a 1423 = a 1432 = − 1 , a 2134 = a 2143 = a 2314 = a 2341 = a 2413 = a 2431 = − 1 , a 3124 = a 3142 = a 3214 = a 3241 = a 3412 = a 3421 = − 1 , a 4123 = a 4132 = a 4213 = a 4231 = a 4312 = a 4321 = − 1 ,

and other a i 1 i 2 i 3 i 4 = 0 . By calculations, we have

a 4444 = 10 < 11 = Λ 4 ( A )

and

a 1111 ( a 4444 − Λ 4 ( A ) + ∣ a 4111 ∣ ) = − 24 < 0 = Λ 4 ( A ) ∣ a 4111 ∣ .

Therefore, tensor A is not strictly diagonally dominate as defined in [21], or quasidoubly strictly diagonally dominant as defined in [22]. However, all the conditions of Theorem 2 can be satisfied.

Λ 1 ( A ) = 12 , Λ 2 ( A ) = 18 , Λ 3 ( A ) = 15 , Λ 4 ( A ) = 11 ,

so N 1 = ∅ , N 2 = { 4 } , N 3 = { 1 , 2 , 3 } . By calculations, we have

w 4 = 11 21 = w , r = 6 11 , P 1 , r ( A ) = 102 11 , P 2 , r ( A ) = 183 11 , P 3 , r ( A ) = 150 11 , r 1 = 132 257 , μ = 11 21 , P 1 , r 1 ( A ) = 1974 257 , P 2 , r 1 ( A ) = 4071 257 , P 3 , r 1 ( A ) = 3300 257 , h = 2827 4137 .

When i = 4 , we obtain

μ ∑ i 2 i 3 i 4 ∈ N 0 3 ∣ a 4 i 2 i 3 i 4 ∣ + ∑ i 2 i 3 i 4 ∈ N 2 3 δ 4 i 2 i 3 i 4 = 0 ∣ a 4 i 2 i 3 i 4 ∣ w + h ∑ i 2 i 3 i 4 ∈ N 3 3 max j ∈ { i 2 , i 3 , i 4 } P j , r 1 ( A ) ∣ a j j j j ∣ ∣ a 4 i 2 i 3 i 4 ∣ = 11 21 × 0 + 11 21 × 0 + 2827 4137 × 132 257 × 11 = 5324 1379 < 121 21 = 11 × 11 21 = ∣ a 4444 ∣ × w 4 .

So, A is positive definite by Theorem 5, that is, f ( x ) is positive definite.

4 Conclusions

In this paper, we introduced some new criterions for identifying ℋ -tensors, which only depended on the elements of tensor. They were well used to check the positive definiteness of an even-order homogeneous polynomial form f ( x ) ≡ A x m . We also verified the effectiveness of new criteria by several numerical examples.

Acknowledgements

The authors would like to express their cordial gratitude to the referee for valuable comments which improved the paper.

Funding information: This work was supported by the Foundation of Science and Technology Department of Guizhou Province (20191161, 20181079), the Foundation of Education Department of Guizhou Province (2018143) and the Research Foundation of Guizhou Minzu University (2019YB08).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2021-06-26

Revised: 2022-01-24

Accepted: 2022-04-08

Published Online: 2022-06-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2022-0043

Keywords for this article

homogeneous multivariate form; positive definiteness; ℋ-tensors; diagonally dominant tensors; irreducible; non-zero element chain

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