Bounds For The Z-spectral Radius Of Nonnegative Tensors

Tensor eigenvalue problem is one of important research topics in tensor theory.In this dissertation,we mainly consider the properties of Z-eigenpair of an irreducible nonnegative tensor,and calculate a sharper bound for Z-spectral radius of an irreducible and weakly symmetric nonnegative tensor.First of all,we give a introduction which mainly introduces the historical background and theoretical significance of eigenvalues and eigenvectors of higher-order tensors,the practical application and research status of Z-eigenpairs are discussed,and further explain the majority of the work of this paper.Secondly,we introduce some basic theoretical knowledge of tensors,and give the concepts of Z-eigenvalue and Z-eigenvector of tensors,irreducible nonnegative tensors,weakly symmetric tensors.Perron-Frobenious theorem of tensor is elaborated in detail.An upper bound and a lower bound of the positive Z-eigenvalue of the nonnegative tensor are calculated,and finally an upper bound of the ratio of the smallest and largest components of a positive Z-eigenvector for a nonnegative tensor is obtained.Finally,by the ratio of the smallest and largest components of a positive Z-eigenvector for a nonnegative tensor,we present some bounds for the eigenvector and Z-spectral radius of an irreducible and weakly symmetric nonnegative tensor.The proposed bounds complement and extend some existing results.And several examples are given to show that such a bound is different from one given in the literature.