Generalized Inverse Theory Of Third-order Tensors Based On The T-product

In this dissertation,we mainly study generalized inverse of tensors,generalized CayleyHamilton theorem and characterization of some special tensors based on the T-product.This dissertation is divided into four parts.In the first part,we first introduce the development of generalized inverse theory,EP matrix theory and tensor generalized inverse theory.Then,we introduce the development,application and research status of the T-product.Finally,we introduce the main research content of this dissertation and give symbolic explanation.In the second part,we introduce the basic concepts and lemmas of tensors based on the T-product which lays a foundation for the study of generalized inverse theory of tensors and special tensors.In the third part,we introduce the definition of T-DMP inverse,T-CMP inverse and Tweighted Moore-Penrose inverse of third-order tensors based on the T-product,and give some characterizations and properties of them.In the fourth part,firstly,the Cayley-Hamilton theorem of third-order tensors is extended to T-Drazin inverses and T-DMP inverses.Then,we study the definition and equivalent characterization of third-order T-range Hermitian（T-EP）tensors.Furthermore,the definition and equivalent characterization of T-weighted EP tensors are introduced through T-weighted Moore-Penrose inverse.