Estimation Of H-eigenvalues Of Third-order Tensors

Tensor is a high-order generalization of matrix.Because of its high information carrying capacity,it has become an effective way to express complex data.It is often widely used in hypergraph,medical MRI,control system stability,face recognition and computer vision.As a part of tensor theory,tensor eigenvalue theory has also attracted extensive attention.On the basis of previous studies,this paper estimates the tensor eigenvalues more accurately,which improves the existing theoretical results.The structure of this thesis is as follows:In the first chapter,the origin and research status of the concept of tensor are introduced,and the definition of tensor eigenvalue and other basic definitions and symbols are given.In the second chapter,H-eigenvalues of the third-order tensors are transformed into eigenvalue problems of symmetric matrices.The upper and lower bounds of the third-order tensor H-eigenvalues are given by using the bounds of the matrix eigenvalues.Then a lower bound on the maximum H-eigenvalue of the third-order tensor is given based on matrix elements.Numerical examples show that-in some cases the bounds obtained by these two methods are better than the existing results.Finally,based on the upper bound of the obtained H-eigenvalues,a sufficient condition for the third-order tensor to be a nonsingular tensor is given.Further,some sufficient conditions for the existence and uniqueness of positive solutions of tensor complementarity problems and nonhomogeneous linear equations are given.In the thirdth chapter,we avoid solving the matrix eigenvalues.Based on the upper and lower bounds of the H-eigenvalues of the third-order tensors given in the second chapter,we obtain the bounds of the H-eigenvalues of the third-order tensor based on the matrix elements by means of the disk theorem.Finally,the application of lower bound of H-eigenvalues in multilinear systems are given.In the fourth chapter,we summarize the article and put forward the feasibility analysis of the next research topic.