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Inequalities for M-tensors
Journal of Inequalities and Applications volume 2014, Article number: 114 (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.
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 [1–7].
If there are a complex number λ and a nonzero complex vector x that are solutions of the following homogeneous polynomial equations:
then λ is called the eigenvalue of and x the eigenvector of associated with λ, where and are vectors, whose i th component is
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 [10–12], which are generalized from nonnegative matrices.
Definition 1.1 The tensor is called reducible if there exists a nonempty proper index subset such that , , . If is not reducible, then we call to be irreducible.
Let , where denotes the modulus of λ. We call the spectral radius of tensor .
Theorem 1.2 If is irreducible and nonnegative, then there exists a number and a vector such that . Moreover, if λ is an eigenvalue with a nonnegative eigenvector, then . If λ is an eigenvalue of , then .
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 ℬ and a real number , where ℬ is the spectral radius of ℬ, such that
Theorem 1.4 Let be an M-tensor and denote by the minimal value of the real part of all eigenvalues of . Then is an eigenvalue of with a nonnegative eigenvector. Moreover, there exist a nonnegative tensor ℬ and a real number such that . If is irreducible, then is the unique eigenvalue with a positive eigenvector.
In this paper, let , we define the i th row sum of as , and denote the largest and the smallest row sums of by
Furthermore, a real tensor of order m dimension n is called the unit tensor, if its entries are for , where
And we define as the set of all the eigenvalues of and
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
Proof Let be an eigenvector of corresponding to , i.e., . For each , we can get
then
Assume that is the smallest component of x,
That is,
so
Similarly, if we assume that , then we can get
Thus, we complete the proof. □
Theorem 2.2 Let be an irreducible M-tensor. Then
where
Proof Because is an eigenvalue of , from Theorem 2.1 in [15], there are , , such that
From Theorem 2.1, we can get
equivalently,
Then, solving for ,
Let be an eigenvector of corresponding to , i.e., , is the smallest component of x. For each , , we can get
Multiplying equations (4) and (5), we get
Then, solving for ,
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 of order 4 dimension 2 with entries defined as follows:
other . By Theorem 1.1 in [15], we have
By Theorem 2.1, we have
By Theorem 2.2, we have
In fact, . 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
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
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 .
Proof Let x be the unique positive eigenvector corresponding to , i.e.,
Let D be the diagonal matrix such that its diagonal entries are components of x, let us check the tensor . It is clear that for ,
Hence 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 . □
Theorem 3.3 Let be an order m dimension n stochastic irreducible nonnegative tensor, , . Then
Proof Let λ be an eigenvalue of the stochastic irreducible nonnegative tensor , x is the eigenvector corresponding to λ, i.e.,
Assume that , then we can get
Then
and therefore,
Thus, we complete the proof. □
Theorem 3.4 Let be an order m dimension n irreducible M-tensor, , . Then
Proof From Theorem 3.2, we may evidently take , and after performing a similarity transformation with a positive diagonal matrix, we may assume that is stochastic. Then, for , the matrix is irreducible nonnegative stochastic, by Theorem 3.3, if , , we can get
That is,
Then
Transforming back to , we get
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 ℤ the set of m-order and n-dimensional real tensors whose off-diagonal entries are nonpositive.
Theorem 4.1 Let , assume that is an M-tensor and . Then ℬ is an M-tensor, and
Proof If , from assume that is an M-tensor and condition (D4) in [14], we know
Because , we can get
then ℬ is an M-tensor.
Let , from Theorem 3.1 and Corollary 3.2 in [13], assume that
where , are nonnegative tensors.
Because and , then we can get
From Lemma 3.5 in [12], we can get
Therefore,
Thus, we complete the proof. □
Theorem 4.2 Let , ℬ be of order m dimension n, suppose that ℬ is an M-tensor and for all . Then, for any eigenvalue λ of , there exists such that .
Proof We first suppose that ℬ is an M-tensor, is an eigenvalue of ℬ with a positive corresponding eigenvector v. Denote
where is the i th component of v. Let
and let λ be an eigenvalue of with x, a corresponding eigenvector, i.e., . Then, as in the proof of Theorem 4.1 in [12], we have
By the definition of , we have , . Applying the first conclusion of Theorem 6 of [8], we can get
Thus, we complete the proof. □
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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.
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He, J., Huang, TZ. Inequalities for M-tensors. J Inequal Appl 2014, 114 (2014). https://doi.org/10.1186/1029-242X-2014-114
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DOI: https://doi.org/10.1186/1029-242X-2014-114