一个新的张量S-型eigenvalue局部化集及其应用
A new S-type eigenvalue localization set for tensors and its applications
摘要：
通过将$N=\{1,2,\cdots,n\}$分解为不相交的子集$S$及其补子集，得到了一个新的张量\emph{S}型特征值局部化集.证明了这个新集合比Qi（Journal of符号计算40（2005）1302-1324），Li等人（Numer.线性代数应用21（2014）39-50）和Li等人（线性代数应用493（2016）469-483）。作为应用，给出了张量正定和半正定的可检验充分条件。此外，基于这个新集合，本文建立了非负张量谱半径的新上界和弱不可约强\emph{M} -张量的最小\emph{H} -本征值的下界。我们证明，这些界限是尖锐的比李等人（编号。线性代数Appl.21（2014）39-50）以及He和Huang（J. Inequal. 114（2014）2014）。数值例子也说明了这一事实。
英文摘要：
A new \emph{S}-type eigenvalue localization set for tensors is derived by breaking $N=\{1,2,\cdots,n\}$ into disjoint subsets $S$ and its complement. It is proved that this new set is tighter than those presented by Qi (Journal of Symbolic Computation 40 (2005) 1302-1324), Li et al. (Numer. Linear Algebra Appl. 21 (2014) 39-50) and Li et al. (Linear Algebra Appl. 493 (2016) 469-483). As applications, checkable sufficient conditions for the positive definiteness and the positive semi-definiteness of tensors are proposed. Moreover, based on this new set, we establish a new upper bound for the spectral radius of nonnegative tensors and a lower bound for the minimum \emph{H}-eigenvalue of weakly irreducible strong \emph{M}-tensors in this paper. We demonstrate that these bounds are sharper than those obtained by Li et al. (Numer. Linear Algebra Appl. 21 (2014) 39-50) and He and Huang (J. Inequal. Appl. 114 (2014) 2014). Numerical examples are also given to illustrate this fact.
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