Sharp Z-Eigenvalue Inclusion Set-Based Method for Testing the Positive Definiteness of Multivariate Homogeneous Forms

Abstract

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.