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In this paper, we establish a sharp Z-eigenvalue inclusion set for even-order real tensors by Zidentity 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.

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