- Open Access
Some inequalities for nonnegative tensors
© He et al.; licensee Springer. 2014
- Received: 26 March 2014
- Accepted: 14 August 2014
- Published: 3 September 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.
- Perron vector
- nonnegative tensor
- spectral radius
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.
The tensor is called nonnegative (or positive) if all the entries (or ). We also denote by ℂ the field of complex numbers.
where denotes the modulus of λ. We call the spectral radius of tensor .
If is not reducible, then we call irreducible.
Theorem 1.3 
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 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 
The result follows. □
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 .
This completes the proof. □
In the following corollary, we get some bounds for and in terms of the entries of and the spectral radius .
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 .
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 .
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.
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.
- 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. 10.1137/100807120MathSciNetView ArticleMATHGoogle Scholar
- Qi L: Eigenvalues and invariants of tensor. J. Math. Anal. Appl. 2007, 325: 1363-1377. 10.1016/j.jmaa.2006.02.071MathSciNetView ArticleMATHGoogle Scholar
- Ng M, Qi L, Zhou G: Finding the largest eigenvalue of a non-negative tensor. SIAM J. Matrix Anal. Appl. 2009, 31: 1090-1099.MathSciNetView ArticleMATHGoogle Scholar
- Chang KC, Zhang T: Perron-Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 2008, 6: 507-520. 10.4310/CMS.2008.v6.n2.a12MathSciNetView ArticleMATHGoogle Scholar
- 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.2008.09.067MathSciNetView ArticleMATHGoogle Scholar
- 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.2010.06.002MathSciNetView ArticleMATHGoogle Scholar
- 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.2011.06.038MathSciNetView ArticleMATHGoogle Scholar
- Qi L: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 2005, 40: 1302-1324. 10.1016/j.jsc.2005.05.007View ArticleMathSciNetMATHGoogle Scholar
- 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 05Google Scholar
- Yang Y, Yang Q: Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 2010, 31: 2517-2530. 10.1137/090778766MathSciNetView ArticleMATHGoogle Scholar
- Yang Y, Yang Q: Further results for Perron-Frobenius theorem for nonnegative tensors II. SIAM J. Matrix Anal. Appl. 2011, 32: 1236-1250. 10.1137/100813671MathSciNetView ArticleMATHGoogle Scholar
- 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/tensorGoogle Scholar
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