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Sharp Z-eigenvalue inclusion set-based method for testing the positive definiteness of multivariate homogeneous forms

Wang Gang (School of Management Science, Qufu Normal University, Rizhao Shandong, China), wgglj1977@163.com
Sun Linxuan (School of Management Science, Qufu Normal University, Rizhao Shandong, China), slxsx2019@163.com
Wang Yiju (School of Management Science, Qufu Normal University, Rizhao Shandong, China), wyijumail@163.com

In this paper, we establish a sharp Z-eigenvalue inclusion set for even-order real tensors by Z-identity tensor and prove that new Z-eigenvalue inclusion set is sharper than existing results. We propose some sufficient conditions for testing the positive definiteness of multivariate homogeneous forms via new Z-eigenvalue inclusion set. Further, we establish upper bounds on the Z-spectral radius of weakly symmetric nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms. The given numerical experiments show the validity of our results.

Keywords: Z-eigenvalue inclusion set, positive definiteness, Z-identity tensor