Note on $ Z $-eigenvalue inclusion theorems for tensors

Chaoqian Li; Yajun Liu; Yaotang Li

doi:10.3934/jimo.2019129

Article Contents

2021, Volume 17, Issue 2: 687-693. Doi: 10.3934/jimo.2019129

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Note on $ Z $-eigenvalue inclusion theorems for tensors

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

Received: January 2019

Revised: April 2019

Early access: October 2019

Published: March 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 15A18, 15A69, 65F15, 65F10.

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