Further Results for $Z$-Eigenvalue Localization Theorem for Higher-Order Tensors and Their Applications

Abstract

In this paper, we present some new $Z$ -eigenvalue inclusion theorem for tensors by categorizing the entries of tensors, and prove that these sets are more precise than existing results. On this basis, some lower and upper bounds for the $Z$ -spectral radius of weakly symmetric nonnegative tensors are proposed, which improves some of the existing results. As applications, we give some estimates of the best rank-one approximation rate in weakly symmetric nonnegative tensors and the maximal orthogonal rank of real orthogonal tensors, and our results are more precise than existing result in some situations. In particular, for a given symmetric multipartite pure state with nonnegative amplitudes in real field, some theoretical lower and upper bounds for the geometric measure of entanglement are also derived in terms of the bounds for $Z$ -spectral radius. Numerical examples are given to illustrate validity and superiority of our results.