Research On Some Problems Of Generalized Inverse Of Tensors

Recently,in many papers,Scholars have extended some concepts and properties of generalized inverses from matrix space to even-order tensor space.For example,Moore-Penrose inverse of tensor,weighted Moore-Penrose inverse and Drazin in-verse,etc.The generalized inverse of tensor is widely used,one of which is that it can be applied to tensor equation to express the solution of some kind of tensor equation.In the process of solving tensor equation,it is an important research topic to study the error between the real solution and the computational solution,so it is necessary to study the perturbation of the corresponding generalized inverse.At the beginning of this paper,we introduce the rank of tensor,2-norm of tensor and the inner product,and prove some related results.Secondly,we further study the results weighted norm of tensor,and give the relationship between weighted norm of tensor and the generalized singular value by using generalized singular value decomposition of tensor,the decomposition formula of perturbation for weighted Moore-Penrose inverse of tensor is given.The perturbation of tensor weighted Moore-Penrose inverse is analyzed by using this decomposition and rank-preserving condition,and the perturbation theorem is improved under some similar conditions,and the condition number of weighted Moore-Penrose inverse is given.Finally,the definitions and lemmas of the Drazin inverse of tensor are given,and then the perturbation of the Drazin inverse of tensor is analyzed by using Frobenius norm of tensor.The application of the perturbation theorem of the Drazin inverse of tensor in perturbation equation is studied.