S-type Localization Sets Of Eigenvalues For Tensors And Its Applications

As the generalization of vectors and matrices,tensors can be used to describe the properties of practical problems more precise,especially in expressing the multi-dimensional indictor data.Many problems in magnetic resonance higher order diffu-sion tensor imaging,on-line face recognition,data mining and processing,computer vision,neural network,nonlinear optimization,high order Markov chain,quantum entanglement,stoichiometry and psychological and so on all have a close linked with tensors.And as an important part of the theory of tensors,tensors eigenvalues and positive definiteness of tensors play an important role in theoretical research and real applications.In this dissertation,we main study the localization of tensors eigenvalues and its applications in identifying the positive definiteness of tensors.Firstly,two S-type localization sets of eigenvalues for tensors are given,and the relations among the proposed results and the Gersgorin-type eigenvalue localization set presented by Qi as well the Brauer-type eigenvalue localization set derived by Li et al.are established.Furthermore,by using these S-type localization sets of tensors eigen-values,two tighter upper bounds for the spectral radius of non-negative tensors have obtained.Secondly,by using the relation between the localization sets of tensors eigenvalues and the class of diagonally dominant tensors,three types of new diago-nally dominant structure tensors are introduced,are applied to identify the positive definiteness of tensors and some sufficient conditions for positive definiteness of even order real symmetric tensors are given.Three algorithms for identifying the posi-tive definiteness of tensors are obtained by programming these sufficient conditions Thirdly,three types of new B-type structure tensors are obtained by combining the technique of tensors decomposition with the proposed three new types of diagonally dominant tensors.Finally,these B-type structure tensors are applied to identify the positive definiteness of tensors and to locate the H-eigenvalues of even order real symmetric tensors,and some new criteria for the positive definiteness of tensors and more precise localization sets for the H-eigenvalues of even order real symmetric tensors are obtained.