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.
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 .
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 .
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.
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