Estimation Of Tensor Eigenvalues

It is well known that tensors are higher-order generalization of matrices and tensor theory is widely used in image processing,data mining,hypergraph theory,continuum mechanics and many other fields.There is a fundamental research problem for the study of tensor theory,namely,eigenvalues of tensors.In this thesis,we investigate estimating of eigenvalues of tensors and improve them compared with some existing results.This thesis is organized as follows:In Chapter 1,we first introduce the research background and the development of eigenvalues of tensors.Then we formally present the definition of eigenvalues of tensors.At last,we add a comment on the notations and definitions that will be used in the sequel.In Chapter 2,given a nonempty proper subset S ofN ,we obtain two Z-eigenvalue inclusion sets which are related to the set for general tensors.The eigenvalues of new inclusion sets are contained in the existing inclusion set.And the results are applied to the estimation of the maximum Z-eigenvalue for a nonnegative tensor.In Chapter 3,we estimate the eigenvalues of nonnegative tensors from two respects.On the one hand,we establish Brauer-type bounds of the Z-spectral radius for weakly symmetric nonnegative tensors based on the properties of Z-spectral radius.As an application of the above result,a upper bound on the largest Z-eigenvalue of the adjacency tensor for uniform hypergraph is presented and a numerical example shows the validity of the proposed bound.On the other hand,we consider the maximal H-eigenvalue of a nonnegative tensor.We obtain the upper and lower bound of the H-spectral radius of a weakly irreducible nonnegative tensor.Meanwhile,we demonstrate that these bounds are tighter than those provided by the literature in theory.In Chapter 4,we briefly conclude the paper with the discussion of some future work.