The Criterion Of H-tensor And Its Application

As an important branch of mathematics,tensor is a higher-order extension of scalars,vectors and matrices.As an extension of H-matrix,H-tensor has the same special structure as H-matrix and plays a vital role in tensor analysis and operation.The nature of tensors,judgment criteria and related issues have become hot topics of concern to many scholars in recent years.In addition,the positive definiteness of evenorder homogeneous polynomials closely related to H-tensors is used in medical image scanning and processing,data analysis and data mining,digital signal processing,stability analysis of nonlinear autonomous systems,quantum entanglement problems,and multivariate neural networks and so on all have a close relation.However,in theory and practical applications,there are often many difficulties in the detection or determination of many structural tensors.Based on this,this thesis will do further research on an important and difficult to determine structure tensor-H-tensor on the basis of related research.At the same time,also according to the consistency of the positive definiteness between the even-order real symmetric tensor and the H-tensor,and further research on the positive definiteness of multivariate even-degree homogeneous polynomials.The specific content is as follows:First of all,this thesis by constructing different positively diagonal matrix transformation factors and gives several practical judgment criteria about H-tensor by using the inequality scaling technique.In view of the failure of the existing judgment criteria,this article The given criterion can still determine that some tensors are H-tensors.At the same time,numerical examples illustrate the effectiveness and superiority of the given criterion.In addition,some sufficient and necessary conditions for H-tensors are also established.As an important application of these criteria,a criterion for judging the positive definiteness of even-order real symmetric tensors is further given,that is,a criterion for judging the positive definiteness of a class of even-order multivariate homogeneous polynomials.In order to better illustrate these facts,some numerical examples are also passed after each criterion to illustrate the feasibility and effectiveness of the results.