Eigenvalue problems of higher order tensors have become an important topic of study in a new applied mathematics branch, numerical multilinear algebra, and they have a wide range of practical applications [2] - [9] .

First, we recall some definitions on tensors. Let ℝ be the real field. An m-th order n dimensional square tensor A consists of nm entries in ℝ , which is defined as follows:

A = ( a i 1 i 2 ⋯ i m ) ,     a i 1 i 2 ⋯ i m ∈ ℝ ,       1 ≤ i 1 , i 2 , ⋯ i m ≤ n .

To an n-vector x, real or complex, we define the n-vector:

A x m − 1 = ( ∑ i 2 , ⋯ , i m = 1 n a i i 2 ⋯ i m x i 2 ⋯ x i m ) 1 ≤ i ≤ n .

and

x [ m − 1 ] = ( x i m − 1 ) 1 ≤ i ≤ n .

If A x m − 1 = λ x [ m − 1 ] , x and λ are all real, then λ is called an H-eigenvalue of A and x an H-eigenvector of A associated with λ [10] [11] .

Qi [10] generalized Geršgorin eigenvalue inclusion theorem from matrices to real supersymmetric tensors, which can be easily extended to generic tensors; see [1] .

Theorem 1. Let A = ( a i 1 i 2 ⋯ i m ) be a complex tensor of order m dimension n . Then

σ ( A ) ⊆ Γ ( A ) = ∪ i ∈ N   Γ i ( A )

where τ ( A ) is the set of all the eigenvalues of A and

Γ i ( A ) = { z ∈ ℂ : | z − a i ⋯ i | ≤ r i ( A ) } ,

where

δ i 1 ⋯ i m = { 1,       if   i 1 = ⋯ = i m 0,           otherwise ,

and

r i ( A ) = ∑ δ i i 2 ⋯ i m = 0 | a i i 2 ⋯ i m | .

Recently, Li et al. [1] obtained the following result, which is also used to identify the positive definiteness of an even-order real supersymmetric tensor.

Theorem 2. Let A = ( a i 1 i 2 ⋯ i m ) be a complex tensor of order m dimension n . Then

σ ( A ) ⊆ K ( A ) = ∪ i , j ∈ N , i ≠ j K i , j ( A )

where σ ( A ) is the set of all the eigenvalues of A and

K i , j ( A ) = { z ∈ ℂ : ( | z − a i ⋯ i | − r i j ( A ) ) | z − a j ⋯ j | ≤ | a i j … j | r j ( A ) } ,

where

r i j ( A ) = ∑ δ i i 2 ⋯ i m = 0 , δ j i 2 ⋯ i m = 0 | a i i 2 ⋯ i m | = r i ( A ) − | a i j ⋯ j | .

In this paper, we give some new eigenvalue localization sets for tensors, which are tighter than those provided by Li et al. [1] .

Theorem 3. Let A = ( a i 1 i 2 ⋯ i m ) be a complex tensor of order m dimension n . Then

σ ( A ) ⊆ Δ ( A ) = ∩ i ∈ N ∪ j ∈ N , j ≠ i Δ i , j ( A )

where σ ( A ) is the set of all the eigenvalues of A and

Δ i , j ( A ) = { z ∈ ℂ : | z − a i ⋯ i | ( | z − a j ⋯ j | − r j i ( A ) ) ≤ | a j i ⋯ i | r i ( A ) } ,

where

r j i ( A ) = ∑ δ j i 2 ⋯ i m = 0 , δ i i 2 ⋯ i m = 0 | a j i 2 ⋯ i m | = r j ( A ) − | a j i ⋯ i | .

Proof. Let x = ( x 1 , ⋯ , x n ) T be an eigenvector of A corresponding to λ ( A ) , that is,

A x m − 1 = λ x [ m − 1 ] . (1)

Let

| x p | = m a x { | x i | , i ∈ N } .

Obviously, | x p | > 0 . For any q ≠ p , from equality (1), we have

| λ − a p ⋯ p | | x p | m − 1 ≤ ∑ δ p i 2 ⋯ i m = 0 | a p i 2 ⋯ i m | | x i 2 | ⋯ | x i m | ≤ ∑ δ q i 2 ⋯ i m = 0 , δ p i 2 ⋯ i m = 0 | a p i 2 ⋯ i m | | x i 2 | ⋯ | x i m | + | a p q ⋯ q | | x q | m − 1 ≤ ∑ δ q i 2 ⋯ i m = 0 , δ p i 2 ⋯ i m = 0 | a p i 2 ⋯ i m | | x p | m − 1 + | a p q ⋯ q | | x q | m − 1 ≤ r p q ( A ) | x p | m − 1 + | a p q ⋯ q | | x q | m − 1 . (2)

That is,

( | λ − a p ⋯ p | − r p q ( A ) ) | x p | m − 1 ≤ | a p q ⋯ q | | x q | m − 1 . (3)

If | x q | = 0 for all q ≠ p , then | λ − a p ⋯ p | − r p q ( A ) ≤ 0 , and λ ∈ Δ ( A ) . If | x q | > 0 , from equality (1), we have

| λ − a q ⋯ q | | x q | m − 1 ≤ r q ( A ) | x p | m − 1 . (4)

Multiplying inequalities (3) with (4), we have

| λ − a q ⋯ q | ( | λ − a p … p | − r p q ( A ) ) ≤ r q ( A ) | a p q ⋯ q | , (5)

which implies that λ ∈ Δ p , q ( A ) . From the arbitrariness of q, we have λ ∈ Δ ( A ) . ,

Remark 1. Obviously, we can get K ( A ) ⊆ Δ ( A ) . That is to say, our new eigenvalue inclusion sets are always tighter than the inclusion sets in Theorem 2.

Remark 2. If the tensor A is nonnegative, from (5), we can get

( λ − a q ⋯ q ) ( λ − a p ⋯ p − r p q ( A ) ) ≤ r q ( A ) a p q ⋯ q .

Then, we can get,

λ ≤ 1 2 { a p ⋯ p + a q ⋯ q + r p q ( A ) + Θ p , q 1 2 ( A ) }

where

Θ p , q ( A ) = ( a p ⋯ p − a q ⋯ q + r p q ( A ) ) 2 + 4 a p q ⋯ q r q ( A ) .

From the arbitrariness of q, we have

λ ≤ m a x i ∈ N m i n j ∈ N , j ≠ i 1 2 { a j ⋯ j + a i ⋯ i + r j i ( A ) + Θ j , i 1 2 ( A ) } .

That is to say, from Theorem 3, we can get another proof of the result in Theorem 13 in [12] .

Jun He is supported by Science and technology Foundation of Guizhou province (Qian ke he Ji Chu [2016]1161); Guizhou province natural science foundation in China (Qian Jiao He KY [2016]255); The doctoral scientific research foundation of Zunyi Normal College (BS [2015]09); High-level innovative talents of Guizhou Province (Zun Ke He Ren Cai [2017]8). Yan-Min Liu is supported by National Natural Science Foundations of China (71461027); Science and technology talent training object of Guizhou province outstanding youth (Qian ke he ren zi [2015]06); Guizhou province natural science foundation in China (Qian Jiao He KY [2014]295); 2013, 2014 and 2015 Zunyi 15,851 talents elite project funding; Zhunyi innovative talent team(Zunyi KH (2015)38). Tian is supported by Guizhou province natural science foundation in China (Qian Jiao He KY [2015]451); Scienceand technology Foundation of Guizhou province (Qian ke he J zi [2015]2147). Xiang-Hu Liu is supported by Guizhou Province Department of Education Fund KY [2015]391, [2016]046; Guizhou Province Department of Education teaching reform project [2015]337; Guizhou Province Science and technology fund (qian ke he ji chu) [2016]1160.

He, J., Liu, Y.M., Tian, J.K. and Liu, X.H. (2017) A Note on the Inclusion Sets for Tensors. Advances in Linear Algebra & Matrix Theory, 7, 67-71. https://doi.org/10.4236/alamt.2017.73006