Brauer-type Localization Sets Of The Pseudo-spectrum Of Tensors

Tensors have important applications in the fields of multivariate polynomial positive definite analysis,asymptotic stability analysis of homogeneous dynamic systems,data mining and processing,image processing,hypergraph theory,etc.Tensor pseudospectra as one mathematical tool is used to describe and study tensor perturbation,and becomes one of the important topics in tensor theory.In this thesis,we study the location problem for the pseudospectra of tensors,obtain a new localization set of the pseudospectra,and proves that the obtained results are better than those in [J.He,C.Q.Li,Y.Wei.Pseudospectra localization sets of tensors with applications.Journal of Computational and Applied Mathematics,2020,369: 1-19].As applications,the measurement for the positive definiteness of even order homogeneous multivariate polynomials are given,and some sufficient conditions for the existence and uniqueness of the solution of the tensor complementarity problem are also given.In particular,for the second-order tensor（matrix）case,based on the spectral norm of matrices and the inequality scaling techniques,two new localization sets of the pseudospectra for matrices are given,and it is proved that the obtained sets are better than those in [M.Embree,L.N.Trefethen.Generalizing eigenvalue theorems to pseudospectra theorems.SIAM Journal on Scientific Computing,2001,23（2）: 583-590].The new sets are applied to analyze the stability of soil ecosystems.