| 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.After that many topics related to the eigenvalues of tensors has attracted much attention,such as nonnegative tensor theory,hypergraph theory and tensor equations.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.In 2016,L.Sun et al.firstly used the generalized inverse of tensors via Einstein product to establish the solutions of tensor equations.In this thesis,we study the properties of the Drazin inverse of tensors and then rank of tensors via Einstein product.The main conclusions are the follows:1.Some basic properties of Drazin inverse of tensors are given.The Cline formula of tensors is established.A necessary and sufficient condition for the existence of the group inverse of tensors via Einstein product are given.2.We define two kinds of ‘block tensors’ including ‘lower triangular tensor’ and‘anti-angular tensor’ by dividing the tensor index sets and the expressions of Drazin inverses of two kinds of ‘block tensors’ are given.In addition,the expression of Drazin inverse of the sum of tensors is given.3.We extend the concepts of matrix series,derivative and integral to tensors.Some properties of the series,derivative and integral of tensors are given.Further,we use the Drazin inverse of tensors to establish the solution of a tensor differential equation via Einstein product.4.The Einstein rank of a tensor is defined by uniting the Einstein product of tensors.For this type of rank of tensors,some inequalities of Einstein rank of tensors are given.At the same time,the relation between Einstein rank of tensors and rank of tensors is considered.For rank-1 tensor,we have the Einstein rank of tensors is 1 as well. |
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