Exclusion sets in eigenvalue inclusion sets for tensors
This is an incremental improvement in eigenvalue localization for tensors, providing tighter inclusion sets for researchers in tensor analysis.
The authors propose two new eigenvalue inclusion sets for tensors by excluding regions that contain no eigenvalues from existing sets. They prove these new sets are subsets of the Geršgorin and Brauer-type sets, and derive sufficient conditions for a tensor to have nonzero determinant.
By excluding some sets, which don't include any eigenvalue of a tensor, from some existing eigenvalue inclusion sets, two new sets are given to locate all eigenvalues of a tensor. And it is shown that these two sets are contained in the Geršgorin eigenvalue inclusion set of tensors provide by Qi (Journal of Symbolic Computation 2005; 40:1302-1324) and the Brauer-type eigenvalue inclusion set provide by Li et al. (Numer. Linear Algebra Appl. 2014; 21:39-50) respectively. Two sufficient conditions such that the determinant of a tensor is not zero are also provided.