Z-eigenvalue Localization And Pseudospectra Localization Of A Tensor

The Z-eigenvalues of a tensor have important applications in the judgement of positive definiteness of even-order homogeneous multivariate polynomials.If all Zeigenvalues of this tensor are greater than zero,then this even homogeneous multivariate polynomial corresponding to this tensor is positive definite.The pseudospectra of a tensor mainly reflects the sensitivity of the eigenvalue of this tensor to perturbation.In this paper,the localization of Z-eigenvalues and the localization of pseudospectra for a tensor and their application are studied,and the following results are obtained:In the aspect of Z-eigenvalue location of a tensor and its applications,a Z-eigenvalue location set with parameters for a sixth-order tensor is given,and then by finding the optimal value of this parameters,the optimal parameter location set is given.As applications,a sufficient condition for the positive definiteness of a sixth-order real symmetric tensor(or a sixth-order homogeneous multivariate polynomial)is given,and then the strong ellipticity conditions of an elasticity tensor and the asymptotic stability of a polynomial systems with delay are judged by this sufficient condition.In the aspect of pseudospectra localization of a tensor and its applications,a pseudospectra localization set and an exclusion set of a tensor are given.As applications,an upper bound of the maximum modulus of the asymptotically stable set is estimated by using the obtained pseudospectra location set,and then the asymptotic decay rate of a homogeneous dynamic system is estimated.In addition,a lower bound for judging the positive definiteness preservation measure of an even homogeneous multivariate polynomial is given,and the sufficient conditions for the global uniqueness and solvability of tensor complementarity problems are given.Finally,the work done in this paper is summarized and the questions for future research are proposed.