Open Access

Inequalities for M-tensors

Journal of Inequalities and Applications20142014:114

https://doi.org/10.1186/1029-242X-2014-114

Received: 3 January 2014

Accepted: 27 February 2014

Published: 13 March 2014

Abstract

In this paper, we establish some important properties of M-tensors. We derive upper and lower bounds for the minimum eigenvalue of M-tensors, bounds for eigenvalues of M-tensors except the minimum eigenvalue are also presented; finally, we give the Ky Fan theorem for M-tensors.

MSC:15A18, 15A69, 65F15, 65F10.

Keywords

M-tensors nonnegative tensor spectral radius eigenvalues

1 Introduction

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 [17].

If there are a complex number λ and a nonzero complex vector x that are solutions of the following homogeneous polynomial equations:
A x m 1 = λ x [ m 1 ] ,
then λ is called the eigenvalue of and x the eigenvector of associated with λ, where A x m 1 and x [ m 1 ] are vectors, whose i th component is
A x m 1 : = ( i 2 , , i m = 1 n a i i 2 i m x i 2 x i n ) 1 i n , x [ m 1 ] : = ( x i m 1 ) 1 i n .

This definition was introduced by Qi and Lim [8, 9] where they supposed that is an order m dimension n symmetric tensor and m is even. First, we introduce some results of nonnegative tensors [1012], which are generalized from nonnegative matrices.

Definition 1.1 The tensor is called reducible if there exists a nonempty proper index subset J { 1 , 2 , , n } such that a i 1 , i 2 , , i m = 0 , i 1 J , i 2 , , i m J . If is not reducible, then we call to be irreducible.

Let ρ ( A ) = max { | λ | : λ  is an eigenvalue of  A } , where | λ | denotes the modulus of λ. We call ρ ( A ) the spectral radius of tensor .

Theorem 1.2 If is irreducible and nonnegative, then there exists a number ρ ( A ) > 0 and a vector x 0 > 0 such that A x 0 m 1 = ρ ( A ) x 0 [ m 1 ] . Moreover, if λ is an eigenvalue with a nonnegative eigenvector, then λ = ρ ( A ) . If λ is an eigenvalue of , then | λ | ρ ( A ) .

The authors in [13, 14] extended the notion of M-matrices to higher-order tensors and introduced the definition of an M-tensor.

Definition 1.3 Let be an m-order and n-dimensional tensor. is called an M-tensor if there exist a nonnegative tensor and a real number c > ρ ( B ) , where is the spectral radius of , such that
A = c I B .

Theorem 1.4 Let be an M-tensor and denote by τ ( A ) the minimal value of the real part of all eigenvalues of . Then τ ( A ) > 0 is an eigenvalue of with a nonnegative eigenvector. Moreover, there exist a nonnegative tensor and a real number c > ρ ( B ) such that A = c I B . If is irreducible, then τ ( A ) is the unique eigenvalue with a positive eigenvector.

In this paper, let N = { 1 , 2 , , n } , we define the i th row sum of as R i ( A ) = i 2 , , i m = 1 n a i i 2 i m , and denote the largest and the smallest row sums of by
R max ( A ) = max i = 1 , , n R i ( A ) , R min ( A ) = min i = 1 , , n R i ( A ) .
Furthermore, a real tensor of order m dimension n is called the unit tensor, if its entries are δ i 1 i m for i 1 , , i m N , where
δ i 1 i m = { 1 , if  i 1 = = i m , 0 , otherwise .
And we define σ ( A ) as the set of all the eigenvalues of and
r i ( A ) = δ i i 2 i m = 0 | a i i 2 i m | , 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 continue this research on the eigenvalue problems for tensors. In Section 2, some bounds for the minimum eigenvalue of M-tensors are obtained, and proved to be tighter than those in Theorem 1.1 in [15]. In Section 3, some bounds for eigenvalues of M-tensors except the minimum eigenvalue are given. Moreover, the Ky Fan theorem for M-tensors is presented in Section 4.

2 Bounds for the minimum eigenvalue of M-tensors

Theorem 2.1 Let be an irreducible M-tensor. Then
τ ( A ) min { a i i } ,
(1)
R min ( A ) τ ( A ) R max ( A ) .
(2)
Proof Let x > 0 be an eigenvector of corresponding to τ ( A ) , i.e., A x m 1 = τ ( A ) x [ m 1 ] . For each i N , we can get
( a i i τ ( A ) ) x i m 1 = δ i i 2 i m = 0 a i i 2 i m x i 2 x i m 0 ,
then
τ ( A ) min { a i i } .
Assume that x s is the smallest component of x,
( a s s τ ( A ) ) x s m 1 = δ s i 2 i m = 0 a s i 2 i m x i 2 x i m 0 .
That is,
τ ( A ) δ s i 2 i m = 0 a s i 2 i m + a s s ,
so
τ ( A ) R max ( A ) .
Similarly, if we assume that x t = { max x i , i N } , then we can get
τ ( A ) δ t i 2 i m = 0 a t i 2 i m + a t t R min ( A ) .

Thus, we complete the proof. □

Theorem 2.2 Let be an irreducible M-tensor. Then
min i , j N , j i 1 2 { a i i + a j j r i j ( A ) Δ i , j 1 2 ( A ) } τ ( A ) max i , j N , j i 1 2 { a i i + a j j r i j ( A ) Δ i , j 1 2 ( A ) } ,
(3)
where
Δ i , j ( A ) = ( a i i a j j + r i j ( A ) ) 2 4 a i j j r j ( A ) .
Proof Because τ ( A ) is an eigenvalue of , from Theorem 2.1 in [15], there are i , j N , j i , such that
( | τ ( A ) a i i | r i j ( A ) ) | τ ( A ) a j j | | a i j j | r j ( A ) .
From Theorem 2.1, we can get
( a i i τ ( A ) r i j ( A ) ) ( a j j τ ( A ) ) a i j j r j ( A ) ,
equivalently,
τ ( A ) 2 ( a i i + a j j r i j ( A ) ) τ ( A ) + a j j ( a i i r i j ( A ) ) + a i j j r j ( A ) 0 .
Then, solving for τ ( A ) ,
τ ( A ) 1 2 { a i i + a j j r i j ( A ) Δ i , j 1 2 ( A ) } min i , j N , j i 1 2 { a i i + a j j r i j ( A ) Δ i , j 1 2 ( A ) } .
Let x > 0 be an eigenvector of corresponding to τ ( A ) , i.e., A x m 1 = τ ( A ) x [ m 1 ] , x s is the smallest component of x. For each s , t N , s t , we can get
( a t t τ ( A ) ) x t m 1 = δ t i 2 i m = 0 a t i 2 i m x i 2 x i m r t ( A ) x s m 1 ,
(4)
( a s s τ ( A ) ) x s m 1 = δ t i 2 i m = 0 , δ s i 2 i m = 0 a t i 2 i m x i 2 x i m a s t t x t m 1 r t s ( A ) x s m 1 a s t t x t m 1 , ( a s s τ ( A ) r t s ( A ) ) x s m 1 a s t t x t m 1 .
(5)
Multiplying equations (4) and (5), we get
( a t t τ ( A ) ) ( a s s τ ( A ) r t s ( A ) ) a s t t r t ( A ) .
Then, solving for τ ( A ) ,
τ ( A ) 1 2 { a t t + a s s r t s ( A ) Δ t , s 1 2 ( A ) } max i , j N , j i 1 2 { a i i + a j j r i j ( A ) Δ i , j 1 2 ( A ) } .

Thus, we complete the proof. □

We now show that the bounds in Theorem 2.2 are tight and sharper than those in Theorem 1.1 in [15] by the following example. Consider the M-tensor A = ( a i j k l ) of order 4 dimension 2 with entries defined as follows:
a 1111 = 3 , a 1222 = 1 , a 2111 = 2 , a 2222 = 2 ,
other a i j k l = 0 . By Theorem 1.1 in [15], we have
2 τ ( A ) 4 .
By Theorem 2.1, we have
0 τ ( A ) 2 .
By Theorem 2.2, we have
1 2 ( 5 17 ) τ ( A ) 1 2 ( 5 5 ) .

In fact, τ ( A ) = 1 . Hence, the bounds in Theorem 2.2 are tight and sharper than those in Theorem 1.1 in [15].

3 Bounds for eigenvalues of M-tensors except the minimum eigenvalue

In this section, we introduce the stochastic M-tensor, which is a generalization of the nonnegative stochastic tensor.

Definition 3.1 An M-tensor of order m dimension n is called stochastic provided
R i ( A ) = i 2 , , i m = 1 n a i i 2 i m 1 , i = 1 , , n .

Obviously, when is a stochastic M-tensor, 1 is the minimum eigenvalue of and e is an eigenvector corresponding to 1, where e is an all-ones vector.

Theorem 3.2 Let be an order m dimension n irreducible M-tensor. Then there exists a diagonal matrix D with positive main diagonal entries such that
τ ( A ) B = A D ( 1 m ) D D m 1 ,

where B is a stochastic irreducible M-tensor. Furthermore, B is unique, and the diagonal entries of D are exactly the components of the unique positive eigenvector corresponding to τ ( A ) .

Proof Let x be the unique positive eigenvector corresponding to τ ( A ) , i.e.,
A x m 1 = τ ( A ) x [ m 1 ] .
Let D be the diagonal matrix such that its diagonal entries are components of x, let us check the tensor C = A D ( 1 m ) D D . It is clear that for i = 1 , 2 , , n ,
i 2 , , i m = 1 n C i i 2 i m = ( C e m 1 ) i = ( A D ( 1 m ) D D m 1 e m 1 ) i = τ ( A ) .

Hence B = C / τ ( A ) is the desired stochastic M-tensor. Since the positive eigenvector is unique, then B is unique, and the diagonal entries of D are exactly the components of the unique positive eigenvector corresponding to τ ( A ) . □

Theorem 3.3 Let be an order m dimension n stochastic irreducible nonnegative tensor, ω = min a i i , λ σ ( A ) . Then
| λ ω | 1 ω .
Proof Let λ be an eigenvalue of the stochastic irreducible nonnegative tensor , x is the eigenvector corresponding to λ, i.e.,
A x m 1 = λ x [ m 1 ] .
Assume that 0 < | x s | = max i | x i | , then we can get
( λ a s s ) x s m 1 = δ s i 2 i m = 0 a s i 2 i m x i 2 x i m .
Then
| λ a s s | δ s i 2 i m = 0 a s i 2 i m = r s ( A ) = 1 a s s ,
and therefore,
| λ ω | | λ a s s + a s s ω | | λ a s s | + | a s s ω | ( 1 a s s ) + ( a s s ω ) = 1 ω .
(6)

Thus, we complete the proof. □

Theorem 3.4 Let be an order m dimension n irreducible M-tensor, Ω = max a i i , λ σ ( A ) . Then
| Ω λ | Ω τ ( A ) .
Proof From Theorem 3.2, we may evidently take τ ( A ) = 1 , and after performing a similarity transformation with a positive diagonal matrix, we may assume that is stochastic. Then, for θ ( 0 , 1 ) , the matrix A ( θ ) = ( 1 + θ ) I θ A is irreducible nonnegative stochastic, by Theorem 3.3, if λ ( θ ) σ ( A ( θ ) ) , ω ( θ ) = min a i i ( θ ) , we can get
| λ ( θ ) ω ( θ ) | 1 ω ( θ ) .
That is,
| 1 + θ θ λ ( 1 + θ θ max a i i ) | 1 ( 1 + θ θ max a i i ) .
Then
| Ω λ | Ω 1 .
Transforming back to , we get
| Ω λ | Ω τ ( A ) .

Thus, we complete the proof. □

4 Ky Fan theorem for M-tensors

In this section we give the Ky Fan theorem for M-tensors. Denote by the set of m-order and n-dimensional real tensors whose off-diagonal entries are nonpositive.

Theorem 4.1 Let A , B Z , assume that is an M-tensor and B A . Then is an M-tensor, and
τ ( A ) τ ( B ) .
Proof If x > 0 , from assume that is an M-tensor and condition (D4) in [14], we know
A x m 1 > 0 .
Because B A , we can get
B x m 1 A x m 1 > 0 ,

then is an M-tensor.

Let a = max 1 i n B i i , from Theorem 3.1 and Corollary 3.2 in [13], assume that
B = a I CB , A = a I CA ,

where CA , CB are nonnegative tensors.

Because A , B Z and B A , then we can get
CA CB .
From Lemma 3.5 in [12], we can get
ρ ( CA ) ρ ( CB ) .
Therefore,
τ ( A ) τ ( B ) .

Thus, we complete the proof. □

Theorem 4.2 Let , be of order m dimension n, suppose that is an M-tensor and | b i 1 i m | | a i 1 i m | for all i 1 i m . Then, for any eigenvalue λ of , there exists i 1 , , n such that | λ a i i | b i i τ ( B ) .

Proof We first suppose that is an M-tensor, τ ( B ) is an eigenvalue of with a positive corresponding eigenvector v. Denote
W = diag ( v 1 , , v n ) ,
where v i is the i th component of v. Let
C = A W 1 m W W [ m 1 ]
and let λ be an eigenvalue of with x, a corresponding eigenvector, i.e., A x m 1 = λ x [ m 1 ] . Then, as in the proof of Theorem 4.1 in [12], we have
C ( W 1 x ) m 1 = λ ( W 1 x ) m 1 .
By the definition of , we have c i i = a i i , i = 1 , , n . Applying the first conclusion of Theorem 6 of [8], we can get
| λ c i i | δ i i 2 i m = 0 | c i i 2 i m | = v i 1 m | a i i 2 i m | v i 2 v i m v i 1 m | b i i 2 i m | v i 2 v i m = v i 1 m ( b i i v m 1 i 1 , , i m = 1 b i i 2 i m v i 2 v i m ) = b i i τ ( B ) .
(7)

Thus, we complete the proof. □

Declarations

Acknowledgements

This research is supported by NSFC (61170311), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), Sichuan Province Sci. & Tech. Research Project (12ZC1802). The first author is supported by the Fundamental Research Funds for Central Universities.

Authors’ Affiliations

(1)
School of Mathematical Sciences, University of Electronic Science and Technology of China

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Copyright

© He and Huang; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.