Localization For Z-eigenvalues And C-eigenvalues Of Tensors

The Z-eigenvalues of tensors have important applications in the determination of positive definiteness of even homogeneous multivariate polynomials and the best rank-one approximation of weakly symmetric nonnegative tensors.The largest Ceigenvalue of piezoelectric tensors play a major role in the piezoelectric electron effect and inverse electron effect.There is a close relationship between the C-eigenvalues of piezoelectric tensors and the Z-eigenvalues of lifting square tensors.In this paper,the localization(distribution,estimation and calculation)and its application of Z-eigenvalues and C-eigenvalues of tensors are studied.The distribution regions of Z-eigenvalues and C-eigenvalues,upper and lower bounds of largest Zeigenvalues and largest C-eigenvalues,and calculation methods of C-eigenvalues are given.The details are as follows:In chapter 1:We briefly describes the research background and significance of the topic,the research status at home and abroad,and lists the basic concepts,definitions and theorems used in this paper.In chapter 2:Firstly,the inclusion set of Z-eigenvalues of tensors is constructed,and the upper bound of the largest Z-eigenvalues of weakly symmetric nonnegative tensors is given.Secondly,a more accurate Z-eigenvalue localization set is given by constructing Z-eigenvalue exclusion set.Then,a Z-eigenvalue localization set with parameters is constructed,and a sufficient condition for the positive characterization of symmetric tensors of even order is given.Finally,the theoretical results are verified by numerical examples.In chapter 3:Firstly,the relationship between the C-eigenvalue of a piezoelectrictype tensor and the lk,s-singular value of a partial symmetric rectangular tensor is given;Secondly,the C-eigenvalue localization set is constructed,and the upper bound of the largest C-eigenvalue of nonnegative piezoelectric-type tensor is given;Then,the lower bound of the largest C-eigenvalue of the nonnegative piezoelectric-type tensor is given;Then,the relationship between the C-eigenvalue of a piezoelectric-type tensor and the Z-eigenvalue of its lifting square tensor is given,from which the calculation method of C-eigenvalue can be given;Finally,a numerical example is given to verify it.In chapter 4:We summarize the work done in this paper and put forward the future research problems.