Research On The Positioning Of Tensor Eigenvalues

Tensor is a higher-order generalization of matrix,which becomes an effective repre-sentation of complex data due to its high carrying capacity.It is widely used in medical resonance imaging,spectral theory of hypergraph,high order markov,control system sta-bility and other fields.In recent ten years,the research on tensor theory has developed rapidly,and the research on tensor eigenvalue localization has also attracted much atten-tion.In this thesis,based on the previous studies,we make more accurate positioning of the eigenvalues of tensors,and optimize the existing theoretical resultsIn chapter 1,we introduce the research background and current situation of tensor eigenvalue problem at home and abroad,and give the definition of tensor eigenvalue and the lemma related to itIn chapter 2,we break the index set N into disjoint subsets S and its complement,and propose two S type exclusion sets of H-eigenvalue that all the eigenvalues do not belong to them.Furthermore,we establish new H-eigenvalue inclusion sets,which can reduce computations and obtain more accurate numerical results.At the same time,we give two criteria for identifying nonsingular tensorsIn chapter 3,to locate all Z-eigenvalues of a tensor more precisely,we establish three Z-eigenvalue exclusion sets such that all Z-eigenvalues do not belong to them and get three tighter Z-eigenvalue inclusion sets of tensor by using these Z-eigenvalue exclusion sets.Furthermore,we show that the new inclusion sets are tighter than the existing results.In chapeter 4,we propose Brauer-type inequalities and obtain Brauer-type upper bounds on the spectral radius for Hadamard product of two nonnegative tensors based on Brauer-type inclusion sets.Compared among different Brauer-type bounds,we show that these bounds is sharper than the existing results.Finally,the given numerical experiments exhibit the superiority of the bound estimationsIn chapter 5,we briefly conclude the paper with the discussion of some future work.