Determination Of Strong H-tensors And Estimation Of Tensor Eigenvalues

As the extension and development of numerical linear algebra,the numerical mul-tilinear algebra is a new branch of computational mathematics and applied mathemat-ics.In numerical multilinear algebra,the tensor is the main research object,and its related problems have become a hot topic in recent years,especially the determination of structured tensors and the eigenvalue problem of higher order tensors.However,in the theoretical and practical applications,many structured tensors are not always easy to test or identify.Meanwhile,it is also generally difficult to calculate all the eigenval-ues of large and medium-sized tensors accurately.Based on the above facts,we mainly study a class of important structured tensors-strong H-tensor that is not easy to iden-tify,and explore some Ky-Fan type tensor eigenvalue localization sets and the spectral theories for Hadamard product and Fan product of tensors in the thesis.The specific contents are consisted by the following parts:Firstly,we present some new criteria only depending on the elements of the given tensors for judging strong H-tensors which cannot be identified by some existing cri-teria.Moreover,some new necessary and sufficient conditions of strong H-tensors are also provided.As the important applications of these criteria,some sufficient condition-s for identifying the positive definiteness of a class of multivariate forms are obtained.These facts are well illustrated by some numerical examples.Secondly,we propose two iterative algorithms with non-parameter for identifying strong H-tensors,which overcome the drawback of choosing the best value of param-eter in some existing algorithms.In addition,more detailed theoretical analysis of the new proposed iterative algorithms is respectively made.Some numerical experiments are performed to illustrate the feasibility and effectiveness of our algorithms.Thirdly,to locate all the eigenvalues of a given tensor,we give two classes of Ky-Fan type eigenvalue localization sets for tensors(Ky-Fan type eigenvalue localization sets based on nonnegative tensors and Ky-Fan type eigenvalue localization sets based on Z-tensors),which are tighter than those given in some existing literatures,respectively.Under certain conditions,the theoretical comparisons of the new proposed Ky-Fan type eigenvalue localization sets for tensors are established.As the important application-s of the above theories,some sufficient conditions for identifying strong M-tensors,the non-singularity and positive definiteness of tensors are obtained.The correspond-ing numerical examples are given to verify the validity and effectiveness of the main theoretical results.Finally,some new upper bounds on the spectral radius of Hadamard product of nonnegative tensors are given.To show their sharpness,the comparisons among these bounds,including the existing one given by Sun et al.(Some inequalities for the Hadamard product of tensors,Linear Multilinear Algebra 2018,66:1199-1214),are performed.Meanwhile,some important inequalities on the spectral radius of the Hada?mard products of the Hadamard powers for nonnegative matrices are extended to high order tensors.Moreover,we give some lower bounds on the minimum eigenvalue of Fan product of irreducible strong M-tensors,and the sharpness of these lower bounds under different conditions are investigated.Some numerical examples are provided to verify our theoretical results.