Abstract
In this paper some new Z-eigenvalue localization sets for general tensors are established, which are proved to be tighter than those newly derived by Wang et al. [Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 187–198]. Also, some relationships between the Z-eigenvalue inclusion sets presented by Wang et al. and the new Z-eigenvalue localization sets for tensors are given. Besides, we discuss the effects of orthonormal transformations for the proposed sets. As applications of the proposed sets, some improved upper bounds for the Z-spectral radius of weakly symmetric nonnegative tensors are given. Numerical examples are also given to verify the advantages of our proposed results over some known ones.
Highlights
For a positive integer n, let N = {1, 2, . . . , n} and C (R) denote the set of all complex numbers
We continue this research on the Z-spectral radius of weakly symmetric nonnegative tensors and propose some new tighter bounds compared with those in [11, 24, 37, 5] on the basis of the new set derived in this paper
In the following theorem, motivated by [1, Theorem 3.3], we develop another Z-eigenvalue localization set for tensors
Summary
On the basis of the relationship between the Gelfand formula and the spectral radius, Song and Qi [36] developed the new upper bounds for the Z-spectral radius, which improve the ones in [5] After that, He and Huang [11] obtained the Ledermannlike upper bound for the largest Z-eigenvalue of the weakly symmetric positive tensors. Wang et al [37] developed some new upper bounds for the largest Z-eigenvalue of weakly symmetric nonnegative tensors by applying the proposed Z-eigenvalue inclusion sets in [37]. We continue this research on the Z-spectral radius of weakly symmetric nonnegative tensors and propose some new tighter bounds compared with those in [11, 24, 37, 5] on the basis of the new set derived in this paper.
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