The Estimation Of The Eigenvalues Of Tensors And The Generalized Inverse Of Tensors

Tensors have been found widely applications in continuum mechanics and quan-tum mechanics of physics,fluorescence excitation emission of chemistry,communica-tion data of social-network analysis,image compression and face recognition of com-puter vision,hyperspectral image and so on.Plenty of works on tensors have been carried out by lots of researchers.Currently,the spectral theory of tensors is a hot topic.In this article,we will establish an interval containing all the H-eigenvalues of real even order symmetry tensors.We will derive a new characterization of B-tensors,which al-lows us to establish the interval.We call the interval B-interval.Moreover,we introduce new classes of tensors called double B-tensors and quasi-double B-tensors.We estab-lish their relationships with double B-tensors and quasi-double B-tensors and shows that double B-tensors,quasi-double B-tensors are proper subclasses of double B-tensors and quasi-double B-tensors,respectively.Double B-tensors are proper subclasses of quasi-double B-tensors.By using the properties of quasi-double B-tensors,we obtain a new interval,which contains all the H-eigenvalues of real even order symmetry tensors.This interval,called QDB-interval,is tighter than B-interval and is also better than other existing estimation methods in some ways.The Laguerre-Samuelson's inequality in economics is another important work on this article.The inequality shows some relationships among the real observed data and their expectation,variance.Our main contribution is to establish the inequality for complex data by using Bessel's inequality.Meanwhile,we will prove the inequality on probability space.Moreover,we apply these results to locate the eigenvalues of certain matrices and tensors,as well as the complex roots of polynomials.The generalized inverse is an important branch in linear algebra.We will define the Moore-Penrose inverse,{1}-inverses,{1,2}-inverses,{1,3}-inverses and {1,4}-inverses of the tensors.The expressions of the generalized inverses of tensors will also be given by using the general solutions of the tensor equations.Moreover,we investigate the least-squares solutions,the minimal norm solutions and the least-squares minimal nor-m solution of the tensor equations and the relations of them between {1,3}-inverses,{1,4}-inverses and the Moore-Penrose inverse,respectively.An algorithm to compute the Moore-Penrose inverse of an arbitrary tensor is constructed by using the function-s of Matlab.Moreover,we apply the obtained results to higher order Gauss-Markov theorem.Finally,we will discuss the the absorption laws in rings.We will give the equivalent conditions for the absorption laws in terms of the Moore-Penrose,group,core inverse,core inverse dual,{1},{1,2},{1,3},and {1,4} inverses.Moreover,the mixed absorption laws of the generalized inverses are studied.