Z-eigenvalue Localization Sets For Tensors And Its Applications

Tensor Z-eigenvalues are widely used in data analysis,quantum entanglement,signal and image processing,magnetic resonance imaging and other fields.For example,the best rank-one approximation rate in data analysis and the geometric measure of the multi-part pure state entanglement in quantum entanglement can be transformed into finding the limiting Z-eigenvalue,which can be obtained by the tensor Z-eigenvalue inclusion set.Therefore,it is of great scientific significance to study the tensor Z-eigenvalue inclusion set.In this paper,three new Z-eigenvalue inclusion sets of tensors are proposed by decomposes the elements of tensors and proved to be more accurate than some existing inclusion sets.Secondly,for the three Z-eigenvalue inclusion sets,we estimate the Z-spectral radius of weakly symmetric nonnegative tensors.Thirdly,as an application of the Z-spectral radius of weakly symmetric nonnegative tensors,the upper and lower bounds of the best rank-one approximation rate of weakly symmetric nonnegative tensors and the estimates of geometric measures with multi-part pure state quantum entanglement are given.Finally,numerical examples are given to demonstrate the validity of the proposed theory.The first chapter introduces the theoretical background and research status of tensor Z-eigenvalue,and gives the relevant preliminary knowledge,definition and lemma used in this paper.In Chapter 2,we consider the inclusion set of Z-eigenvalues of tensors,and give three new inclusion sets of all Z-eigenvalues of tensors: the first theorem is derived from Ostrowski’s matrix eigenvalue theorem;The second and third theorems are the Z-eigenvalue containing sets obtained by the S-type partition method.In this paper,we prove that the third theorem is more accurate than the second theorem,and the second theorem is more accurate than the existing theorems.Numerical examples show that in some cases,our results are more accurate than the existing tensor Z-eigenvalue inclusions.In Chapter 3,we consider the estimation of the upper and lower bounds of the Z-spectral radius of weakly symmetric nonnegative tensors.In this chapter,we first introduce the results obtained by some scholars.Then we use the Z-eigenvalue inclusion set obtained in Chapter 2 to give some tighter upper and lower bounds of the Z-spectral radius of weakly symmetric nonnegative tensors,improve some existing results,and illustrate the validity of our results by numerical examples.Chapter 4 deals with the application of the upper and lower bounds of theZ-spectral radius of weakly symmetric nonnegative tensors to the best rank-one approximation rate.This chapter first introduces the definition of the best rankone approximation rate and its relationship with the Z-spectral radius of weakly symmetric non-negative tensors.Then,based on the upper and lower bounds of theZ-spectral radius of weakly symmetric non-negative tensors obtained in Chapter 3,as an application,Some upper and lower bounds of the best rank-one approximation rate of tensors are given,and numerical examples are given to show that our results are valid.Chapter 5 deals with the application of Z-radius estimation of weakly symmetric nonnegative tensors to the geometric measures of multi-part pure state quantum entanglement.According to the relation between the geometric measure of pure state entanglement and the tensor Z-spectral radius,the upper and lower bounds of two different definitions of the geometric measure of weakly symmetric pure state entanglement with non-negative amplitude are given,and demonstrate the validity of our results by numerical examples.