Theoretical Analysis Of Eigenvalues And Tensor Equations Of Structural Tensors

Tensor is a high-order extension of vector and matrix,which is widely used in data analysis,nonlinear optimization and hypergraph spectral theory.In this thesis,structural tensor eigenvalue problem,tensor equation and properties of solutions of generalized tensor equations are analyzed.This thesis consists of six chapters.In Chapter 1,we mainly introduce the research background and current situation of tensor eigenvalue problem,tensor equations and generalized tensor equations.In Chapter 2,we first give some basic symbols,then review the concepts of positive definite tensor,semi-positive tensor,irreducible tensor and nonsingular tensor,and give the definition of stable tensor,which makes necessary preparation for the following research.In Chapter 3,we focus on the properties of the eigenvalues of structural tensors.We briefly review the properties of the eigenvalues of diagonally dominant tensor,M-tensor and H-tensor,as well as the discrimination methods of M-tensor and H-tensor.By using the nonnegativity of the real parts of the eigenvalues of diagonally dominant tensor,Mtensor and H-tensor,the relations between them and the semi-positive stable tensor are obtained.In Chapter 4,we mainly study the problem of tensor equations.This chapter is divided into two sections.In the first section,we first study the relationship between strictly diagonally dominant tensor and strong H-tensor and nonsingular tensor,and then prove the nonempty compactness of the solution set of tensor equations when the coefficient tensor is positive semi-definite strong H-tensor.In the second section,the upper and lower bounds of the solutions of tensor equations are discussed under appropriate conditions.In Chapter 5,the existence of solutions to generalized tensor equations is studied.In this section,by using the exceptional clusters of continuous functions and the topological degree theory,we prove the nonempty compactness of the solution set of generalized tensor equations when the first term coefficient tensor is a positive semi-definite strong H-tensor.In Chapter 6,we make a summary of this thesis,and look forward to the future work.