C-S Inverse And WG Matrix

This thesis mainly studies the definition and related properties of C-S inverse,some characterizations of C-S partial order and WG matrix.The full text is divided into four parts.In first part,we mainly introduce the development history of generalized inverse theory,matrix partial order,special matrices and the research status at home and abroad.In second part,we introduce some symbols,basic concepts and lemmas of generalized inverse theory,matrix partial order and special matrices,laying the foundation for the follow-ing research.In the third part,we first introduce a new generalized inverse C-S inverse,and give its equivalent conditions and properties.Then a new binary relation“（?）”is introduced on the basis of C-S inverse.Based on the binary relation,supplementary condition make it a partial order.Furthermore,it gives some characterizations of C-S partial order and examples to show that it is different from other classical partial orders.Finally,some new characterizations of i-EP matrix are given by using C-S inverse.In the fourth part,we use commutativity,matrix decomposition,WG inverse,Drazin inverse and other inverses to give some new characterizations of WG matrix.Furthermore,we generalize the Cayley-Hamilton theorem for special matrices including the WG matrix.Finally,we give examples to verify these results.