# A tighter $Z$-eigenvalue localization set for tensors and its applications

## TL;DR

This paper introduces a new, tighter set for localizing Z-eigenvalues of tensors, improving bounds on spectral radii, with theoretical proofs and numerical validation.

## Contribution

It presents a novel Z-eigenvalue localization set that is proven to be tighter than existing ones, enhancing spectral radius bounds for certain tensors.

## Key findings

- The new localization set is tighter than previous sets.
- A sharper upper bound for the Z-spectral radius is established.
- Numerical examples confirm the theoretical improvements.

## Abstract

A new $Z$-eigenvalue localization set for tensors is given and proved to be tighter than those presented by Wang \emph{et al}. (Discrete and Continuous Dynamical Systems Series B 22(1): 187-198, 2017) and Zhao (J. Inequal. Appl., to appear, 2017). As an application, a sharper upper bound for the $Z$-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.