Fourth-order Tensor Eigenvalue Inclusion Sets And Positive Definiteness Studies

Fourth-order tensors are higher-order generalizations of matrices.It plays an important role in signal processing,wireless communication system,image processing,data analysis and high-order statistics.As an important tool in tensor analysis and calculation,the eigenvalue of tensor has attracted much attention because of its important applications in medical magnetic resonance imaging,data analysis,even order multivariate form of positive definite judgment and other fields,and gradually become a hot topic of research.In this thesis,on the basis of previous studies,we study the inclusion set and positive definiteness of fourth-order tensor eigenvalues,and improve some existing theoretical results.This thesis is organized as follows:In Chapter 1,we first introduce the research background and current situation of tensor eigenvalue problem at home and abroad,then we give some definitions about the eigenvalue of tensor.In Chapter 2,we introduce a Z-identity tensor and establish two Z-eigenvalue inclusion sets for fourth-order tensors,which are sharper than some existing results.As applications,we provide some checkable sufficient conditions for the positive definiteness of fourth-order symmetric tensors.Further,we propose upper bounds on the Z-spectral radius of fourth-order nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms under suitable conditions.In Chapter 3,we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors,which are sharper than some existing results.As applications,we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.In Chapter 4,based on the spectral theory of nonnegative elasticity tensors,we establish an upper bound and sharp lower bounds for the minimum M-eigenvalue of elasticity Z-tensors without irreducible conditions.The obtained results are sharper than some existing results.Numerical examples are given to show the efficiency of the proposed results.In Chapter 5,we briefly conclude the paper with the discussion of some future work.