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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
From the journal Open Mathematics

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.

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

In this paper, we continue to present new criteria for -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 -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 -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 -tensors

In this section, three new criteria for identifying -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 -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 -tensor, then A is an -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 -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 -tensor.

Next, we give some new criteria for -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 -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 = 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 ( ) = 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 ( ) = 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 ( ) = 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 ( ) ( i N ) . So is an -tensor by Lemma 1. From Lemma 2, A is an -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 -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 = 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 ( ) = 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