# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2019129

## Note on $Z$-eigenvalue inclusion theorems for tensors

 School of Mathematics and Statistics, Yunnan University, Kunming 650091, China

Received  January 2019 Revised  April 2019 Published  October 2019

Wang et al. gave four $Z$-eigenvalue inclusion intervals for tensors in [Discrete and Continuous Dynamical Systems Series B, 1 (2017), 187-198]. However, these intervals always include zero, and hence could not be used to identify the positive definiteness of a homogeneous polynomial form. In this note, we present a new $Z$-eigenvalue inclusion interval with parameters for even-order tensors, which not only overcomes the above shortcomings under certain conditions, but also provides a checkable sufficient condition for the positive definiteness of homogeneous polynomial forms, as well as the asymptotically stability of time-invariant polynomial systems.

Citation: Chaoqian Li, Yajun Liu, Yaotang Li. Note on $Z$-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019129
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