Tensor Eigenvalue Estimation And Its Applications

The data analysis and calculation methods based on matrix can not effectively deal with the problem of big data with the progress of science and technology,the drive of big data,the diversity of data sources and types.Therefore,the use of tensor with high-order and high-dimensional structure instead of matrix is bound to greatly affect big data and related research fields.In recent years,the theory,calculation and application of tensor eigenvalue have developed greatly.Tensors eigenvalues have become increasingly significant issue in several diverse fields of applied mathematics and computational mathematics,and promoted the development of numerical multilinear algebra.On the other hand,they have a great diversity of practical applications such as data analysis and mining,signal processing,image processing,magnetic resonance imaging,automatic control,and quantum entanglement.In this paper,we are devoted to tensor eigenvalue estimation and its application,including the C-eigenvalue localization of piezoelectric tensor and its application in piezoelectric materials,H-tensor theory and H-eigenvalue localization,Z-eigenvalue estimation and its application in tensor rank one approximation rate and geometric measurement of quantum entanglement.As follows:(1)We establish a new inclusion set for the C-eigenvalues of piezoelectric tensors,and proved that the set is tighter than the existing result.It is shown that our results are more precise on the grounds of the highest piezoelectric coupling constant of some piezoelectric materials.(2)We introduce exponential type locally generalized strictly double diagonally dominant tensors,and prove that exponential type locally generalized strictly double diagonally dominant tensors must be H-tensor.As applications,some new eigenvalue localization sets and sufficient conditions for the positive definiteness of even-order real symmetric tensors are proposed.(3)For a given m order n dimension tensor,we construct a n ×ndimension tensor-generated matrix,such that the original tensor must be H-tensor when the tensor-generated matrix is H-matrix.We also discuss some similar properties of the original tensor and the tensorgenerated matrix,such as(strong)symmetry,(weak)irreducibility,(strict)diagonal dominance.In addition,based on the relationship between the original tensor and the tensor-generated matrix,we extend some classical results of matrix eigenvalues to tensor eigenvalues,and obtain the modified Brauer set,modified Ostrowski set and modified S-type inclusion set.(4)We present some new Z-eigenvalue inclusion theorem for tensors by categorizing the entries of tensors,and prove that these sets are more precise than existing results.On this basis,some lower and upper bounds for the Z-spectral radius of weakly symmetric nonnegative tensors are proposed.As applications,we give some estimates of the best rank-one approximation rate in weakly symmetric nonnegative tensors and the maximal orthogonal rank of real orthogonal tensors.(5)We are devoted to the geometric measure of entanglement of a multipartite pure state by the means of tensor Z-spectrum theory.On the basis of Chapter 5,we propose some theoretical upper and lower bounds of entanglement for symmetric pure state with nonnegative amplitudes for two kinds of geometric measures with different definitions respectively.