Eigenvalue Inclusion Set Of Structure Tensor And Its Applicatio

Tensors are higher-order generalizations of matrices,many properties of matrices can be generalized to tensors,the research of structured tensors has become an important component of tensor research due to their special structure.With the rise of artificial intelligence and machine learning,the tensor eigenvalue problem has gradually become a hot topic of research.As an important tool in tensor calculation,the eigenvalues of tensors are generalized used in medical magnetic resonance imaging,data analysis,and the determination of positive definiteness of even-order multivariate forms.In this paper,we study the eigenvalue inclusion sets of several structured tensors and their applications.Firstly,we study the bounds for the minimum M-eigenvalue and positive definiteness of elastic Z-tensors.Then we study the bounds for the minimum H~+-singular value of rectangular tensors.Finally,we study the Pareto H-eigenvalue inclusion intervals of the tensor eigenvalue complementarity problem.The main contents are as follows:In chapter 1,we introduce the reserch background and current situation of the tensor eigen-value problem at home and abroad.In chapter 2,by the extreme eigenvalue of symmetric matrices extracted from elastic Z-tensors,we establish an upper bound and three lower bounds for the minimum M-eigenvalue of elastic Z-tensors without irreducible conditions.The obtained results improve the existing results theoretically,and show the validity through numerical examples.As an application,some checkable sufficient and necessary conditions for the positive definiteness of elastic Z-tensors are proposed.In chapter 3,we establish the sharp bounds for the minimum H~+-singular value of rectan-gular tensors.In addition,new sufficient conditions are proposed to identify the copositivity of rectangular tensors.In chapter 4,by dividing the tensor index set into complementary subsets S and S,firstly,we establish an inclusion interval of Pareto H-eigenvalues for the tensor eigenvalue comple-mentarity problem without selecting S;secondly,a new S-type Pareto H-eigenvalue inclusion interval is established,which further reduces the computation.An example shows that the above two inclusion intervals can quite accurately locate Pareto H-eigenvalues.As an applica-tion,some sufficient conditions for testing the strict copositivity of tensors are proposed.In chapter 5,we summarize the main content of this paper and put forward topics that can be studied in the future.