Research On The Spectra Radii Of Tensors And Generalized Inverses Of Tensors

Following the publishing of ‘Methods and Applications of Absolute Differential Geometry' in 1890,a monograph of the Italian mathematician,theoretical physicist,one of the founders of tensor analysis R.Gregorio,tensor is known to many mathematicians.Around 1915,A.Einstein introduced tensors in ‘General Theory of Relativity'.After that,the research of tensors received much attention.Following the rise of quantum computing,machines learn and artificial intelligence,there emerged many new problems in the area of tensor theory,such as,eigenvalues of tensors,spectral theory of hypergraph via tensors and solving tensor equations.In 2005,Professor Liqun Qi from Hong Kong Polytechnic University and Professor Lek-Heng Lim from University of Chicago independently proposed the concept of the eigenvalues of tensors,which has attracted much attention.Tensor equations arise from control systems,theoretical physics and some other engineering problems.Einstein gravitational field equation,three-dimensional viscous flow around equation,and higher order Poisson equation are all tensor equations.Solving tensor equations is an important research problem.The main work of this thesis includes: giving bounds for the spectral radii of nonnegative irreducible tensors;characterizing properties for the spectra of the Hadamard product of tensors;proposing the concepts of the generalized inverses of tensors via the generalized product of tensors and Einstein product,respectively;establishing the solutions of some tensor equations by using the generalized inverses of tensors.The conclusions are given in the following four aspects.1.By using a digraph associated with a tensor,we establish some bounds for the spectral radii of nonnegative weakly irreducible tensors and a Collatz-Wielandt formula via combinatorial methods for the spectral radii.We characterize some bounds for the spectral radii of the adjacency tensor and the signless Laplacian tensor of a uniform hypergraph.2.We give some bounds for the spectra radii of the Hadamard product of tensors.For the minimal value of the real parts of all eigenvalues for the Hadamard product of Z-tensors or M-tensors,we establish some lower bounds and give some criteria of Mtensors.Further,we estimate the spectral radius for the adjacency tensor of a directed weighted hypergraph.3.We define the generalized inverses of tensors via the generalized product of tensors and Einstein product,respectively.And we give the existence for the generalized inverses of tensors and the representations for the generalized inverses of some block tensors.By using the generalized inverses of tensors,we establish the necessary and sufficient conditions and solutions of some tensor equations.