# A new $Z$-eigenvalue inclusion theorem for tensors

## TL;DR

This paper introduces a new, tighter $Z$-eigenvalue inclusion theorem for tensors, providing improved bounds for the $Z$-spectral radius of weakly symmetric nonnegative tensors, supported by numerical examples.

## Contribution

The paper presents a novel $Z$-eigenvalue inclusion theorem that improves upon existing bounds and offers a sharper estimate for the $Z$-spectral radius of certain tensors.

## Key findings

- The new theorem is tighter than previous results.
- A sharper upper bound for the $Z$-spectral radius is established.
- Numerical examples confirm the effectiveness of the proposed bounds.

## Abstract

A new $Z$-eigenvalue inclusion theorem for tensors is given and proved to be tighter than those in [G. Wang, G.L. Zhou, L. Caccetta, $Z$-eigenvalue inclusion theorems for tensors, Discrete and Continuous Dynamical Systems Series B,22(1) (2017) 187--198]. Based on this set, a sharper upper bound for the $Z$-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to show the effectiveness of the proposed bound.