Research On Eigenvalue Inclusion Sets Of Tensors

The concept of tensors was introduced by Guass,Riemann and Christoffel,etc.,in the19th century in the study of differential geometry.It was Einstein who applied tensor in his study of general relativity in 1913.This made tensor become an important tool in theoretical physics,continue mechanics and many other areas of science and engineering.In 2005,Liqun Qi and Lim defined the eigenvalue of tensors,respectively,and some properties for eigenvalues of tensor are discussed.Tensor is a multi-way array and can be regard as the extension of matrix..In 2005,liqun Qi gave Gersgorin-type eigenvalue inclusion sets of tensors.Subsequently,some researchers give a series of results about Gersgorin disk theorem.In 2007,S.M.Cioaba and D.A.Gregory give the principal radio's lower bound of nonnegative irreducible matrices.Since 2008,Zhang Gongqing,etc.,prove that the famous Frobenius-Perron theorem of the nonnegative matrix theory hold for tensors,giving a series of results.As a frontier research subject,the eigenvalues of tensor are faced many problems,which need to be sovled by new methods and new technologies.The main results of this paper as follows:(1)This paper generalizes the Ostrowski Theory to the hyper-symmetric tensor,and get Ostrowski Theorem of hyper-symmetric tensor:(?)(2)The related theorem of boundary point of eigenvalues in Gersgorin disk is also given in this paper,each Gersgorin circle of a tensor goes through its eigenvalues.(3)In 2013,Wang.Z and Wu.W gave the boundary of principal ratio of positive tensor;S.M.Cioaba and D.A.Gregory gave the lower bound of principal ratio of nonnegative matrix.We generalize it to tensor,and get the lower bound of S.M.Cioaba-D.A.Gregory type principal ratio.