Localization And Bounds For Z-eigenvalues Of Tensors And C-eigenvalues Of Piezoelectric-type Tensors

Eigenvalue problems of tensors as one of the important topics in tensor theory are applied wildly to many fields such as the stability of nonlinear systems,high order Markov chains,image matching,polynomial optimization and hypergraph theory.We in this thesis localize or bound two classes of eigenvalues of tensors called Z-eigenvalues and C-eigenvalues,obtain some localization sets for Z-eigenvalues and C-eigenvalues,and some bounds for the Z-spectral radius and the largest C-eigenvalue.Concretely,for localizing Z-eigenvalues of tensors,A new set is given to include all Z-eigenvalues.As applications,a sufficient condition for the positive definiteness for multivariate polynomials and the stability for a class of time-invariant dynamical systems is obtained respectively,and an upper bound for the Z-spectral radius is also given;For bounding Z1-eigenvalues of tensors,we get a new upper bound for the Z1-spectral radius,and prove that the new bound is better than that in[W.Li,D.D.Liu and S.W.Vong.Z-eigenpair bounds for an irreducible nonnegative tensor.Linear Algebra and its Applications,483(2015)182-199]and[G.Y.Wang,C.L.Deng and C.J.Bu.Some upper bounds on eigenvalues of tensors,Applied Mathematics and Computation,329(2018)266-277];For C-eigenvalues of Piezoelectric-type tensors,two intervals for localizing all C-eigenvalues is given,which provide two upper bounds for the largest C-eigenvalue.