Some inequalities for nonnegative tensors
Journal of Inequalities and Applications volume 2014, Article number: 340 (2014)
Let be a nonnegative tensor and its Perron vector. We give lower bounds for and upper bounds for , where and .
MSC:15A18, 15A69, 65F15, 65F10.
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]. The main difficulty in tensor problems is that they are generally nonlinear. Therefore, large amounts of results for matrices are never in force for higher order tensors. However, there are still some results preserved in the case of higher order tensors.
Throughout this paper we consider an m th order n-dimensional tensor consisting of entries in ℝ:
The tensor is called nonnegative (or positive) if all the entries (or ). We also denote by ℂ the field of complex numbers.
Definition 1.1 A pair is called an eigenvalue-eigenvector pair of , if they satisfy
where n-dimensional column vectors and are defined as
where denotes the modulus of λ. We call the spectral radius of tensor .
Definition 1.2 The tensor is called reducible if there exists a nonempty proper index subset such that
If is not reducible, then we call irreducible.
Theorem 1.3 
If is irreducible and nonnegative, then there exist 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 .
We call a Perron vector of corresponding to its largest nonnegative eigenvalue .
In this paper, we are interested in studying some bounds for the Perron vector of . For this purpose, we define
In the following we first give a new and simple bound for . Then we give some lower bounds for and upper bounds for , which can be used to get another bound for .
The paper is organized as follows. In Section 2, some efforts of establishing the bounds of the nonnegative tensor are made. An application of these bounds is studied in Section 3.
In , the authors have studied the perturbation bound for the spectral radius of , and they show that a Perron vector x must be known in advance so that the perturbation bound can be computed. Here we cite a lemma for use below.
Lemma 2.1 
Suppose is nonnegative such that and x is its Perron vector x. Then
Define the i th row sum of as
and denote the largest, the smallest, and the average row sums of by
Theorem 2.2 Suppose is nonnegative with Perron vector x. Then
Proof Since, for any ,
It follows from (4) and (5) that
If we assume , then, for any ,
The result follows. □
Remark By the obvious inequality
we see that our new lower bound in (3) is sharper than that in (2).
If the tensor is positive, we derive a new upper bound for in terms of the entries of and the spectral radius .
Theorem 2.3 If is positive with Perron vector x, then
Proof First, we prove the right side of (9). Now we consider
so we can get
On the other hand,
so we can get
This completes the proof. □
In the following corollary, we get some bounds for and in terms of the entries of and the spectral radius .
Corollary 2.4 If is positive with Perron vector x. Then
Similar to the proof of Theorem 2.3, we can get the results. □
Remark Similar to the proof of Theorem 2.3, we can get the lower bound of .
3 Application to the perturbation bound
Suppose and ℬ is another tensor satisfying . In this section we are interested in the bound of .
Lemma 3.1 
, then .
We let be the positive cone, and let the interior of ℙ be denoted by .
Theorem 3.2 Let be weakly irreducible. For a nonzero , we define
By the result of Lemma 5.4 in , we have
Theorem 3.3 Let , ℬ be irreducible and positive such that and . Then
Proof Let x be the Perron vector of . Define i as follows:
Then, by Theorem 3.2, we have
From the simple equality
and by considering the i th coordinate,
Since and ,
Similarly, we can get
Remark If we let x be the Perron vector of ℬ, then by a similar method, we can get the following bound:
We have obtained a new and sharper bound of , lower bounds for , and upper bounds for , where is the Perron vector of a positive tensor with and . In addition we have given bounds of when by these bounds.
Chang KC, Pearson K, Zhang T: Primitivity, the convergence of the NZQ method, and the largest eigenvalue for nonnegative tensors. SIAM J. Matrix Anal. Appl. 2011, 32: 806-819. 10.1137/100807120
Qi L: Eigenvalues and invariants of tensor. J. Math. Anal. Appl. 2007, 325: 1363-1377. 10.1016/j.jmaa.2006.02.071
Ng M, Qi L, Zhou G: Finding the largest eigenvalue of a non-negative tensor. SIAM J. Matrix Anal. Appl. 2009, 31: 1090-1099.
Chang KC, Zhang T: Perron-Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 2008, 6: 507-520. 10.4310/CMS.2008.v6.n2.a12
Chang KC, Pearson K, Zhang T: On eigenvalue problems of real symmetric tensors. J. Math. Anal. Appl. 2009, 350: 416-422. 10.1016/j.jmaa.2008.09.067
Liu Y, Zhou G, Ibrahim NF: An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor. J. Comput. Appl. Math. 2010, 235: 286-292. 10.1016/j.cam.2010.06.002
Zhou G, Caccetta L, Qi L: Convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor. Linear Algebra Appl. 2013, 438: 959-968. 10.1016/j.laa.2011.06.038
Qi L: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 2005, 40: 1302-1324. 10.1016/j.jsc.2005.05.007
Lim LH: Singular values and eigenvalues of tensors: a variational approach. Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing 2005 CAMSAP 05, 129-132. CAMSAP 05
Yang Y, Yang Q: Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 2010, 31: 2517-2530. 10.1137/090778766
Yang Y, Yang Q: Further results for Perron-Frobenius theorem for nonnegative tensors II. SIAM J. Matrix Anal. Appl. 2011, 32: 1236-1250. 10.1137/100813671
Wen, L, Ng, M: The perturbation bound for the spectral radius of a non-negative tensor. http://www.math.hkbu.edu.hk/mng/tensor-research/tensor
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.
The authors declare that they have no competing interests.
All authors contributed equally to this work. All authors read and approved the final manuscript.
About this article
Cite this article
He, J., Huang, TZ. & Cheng, GH. Some inequalities for nonnegative tensors. J Inequal Appl 2014, 340 (2014). https://doi.org/10.1186/1029-242X-2014-340