Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Z$\end{document}-Eigenvalue Localization Sets for Even Order Tensors and Their Applications

Firstly, a new Geršgorin-type Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Z$\end{document}-eigenvalue localization set with parameters for even order tensors is presented. As an application, some sufficient conditions for the positive (semi-)definiteness of even order real symmetric tensors are obtained. Secondly, by selecting appropriate parameters an optimal set is obtained and proved to be tighter than some existing results. Thirdly, as another application, new upper bounds for the Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Z$\end{document}-spectral radius of even order weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.