Optimal Z-Eigenvalue Inclusion Intervals for Even Order Tensors and Their Applications

Firstly, the optimal Z -eigenvalue inclusion interval for the interval in Theorem 3 of (Acta Appl. Math. 169:323–339, 2020 ) for even order tensors is given. Secondly, by making full use of the information of Z -eigenvectors, new Geršgorin-type Z -eigenvalue inclusion intervals with n parameters for fourth-order and sixth-order tensors are constructed. Thirdly, by selecting appropriate parameters, two optimal intervals for fourth-order and sixth-order tensors are presented. Finally, as applications, several sufficient conditions for the positive definiteness of even order real symmetric tensors (also homogeneous polynomial forms) as well as the asymptotically stability of time-invariant polynomial systems are obtained. Moreover, bounds of the Z -spectral radius of weakly symmetric nonnegative tensors are obtained, which are used to estimate the convergence rate of the greedy rank-one update algorithm and derive bounds of the geometric measure of entanglement of symmetric pure state with nonnegative amplitudes.