Some new inclusion sets for eigenvalues of tensors with application

In this paper, we are concerned with the eigenvalue inclusion sets for tensors. Some new S-type eigenvalue localization sets for tensors are employed by dividing N = {1,2,..., n} into disjoint subsets S and its complement. Our new sets, are proved to be tighter than that newly derived by Huang et al. (J. Inequal. Appl. 2016 (2016) 254). As applications, we can apply the proposed sets for determining the positive (semi-)definiteness of even-order symmetric tensors. Some examples are given to show the sharpness of our new sets in contrast with the known ones, and verify the effectiveness of those in identifying the positive (semi-)definiteness of tensors.

A modified S-type eigenvalue localization set of tensors applications

Based on the S-type eigenvalue localization set developed by Li et al. (Linear Algebra Appl. 493 (2016) 469-483) for tensors, a modified S-type eigenvalue localization set for tensors is established in this paper by excluding some sets from the existing S-type eigenvalue localization set developed by Huang et al. (arXiv: 1602.07568v1, 2016). The proposed set containing all eigenvalues of tensors is much sharper compared with that employed by Li et al. and Huang et al. As its applications, a criteria, which can be utilized for identifying the nonsingularity of tensors, is developed. In addition, we provide new upper and lower bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of weakly irreducible strong M-tensors. These bounds are superior to some previous results, which is illustrated by some numerical examples.