On Strongly Drazin Property

The generalized inverse has been studied by many authors.In this thesis,we studied the properties of strongly Drazin inverse,we shall combine strongly Drazin inverse and strongly nil-clean,and then deduce some properties of strongly Drazin inverse,and apply them to block matrices.The innovation of the thesis chiefly comprises:(1)we study the properties of strongly Drazin inverse,based on the sufficient and necessary conditions of strongly Drazin inverse,and examples are given;(2)we extend Jacobson's lemma and Cline's formula to strongly Drazin inverse,and many polynomial conditions are obtained to make some theorems hold;(3)we apply some theorems of strongly Drazin inverses over rings to complex matrices,and more theorems are applied to block matrix,so that new properties of idempotent-nilpotent decomposition of complex matrices are obtained.This thesis includes the following sections:In chapter 1,we introduce the development of generalized inverse and the background of strongly Drazin inverse and the following work of this thesis.The main results of this thesis are also stated.In chapter 2,we introduce some basic definitions involved in this thesis,and then list some important conclusions in this thesis.In chapter 3,we study strongly Drazin inverse over a ring,we study the Jacobson's lemma and Cline's formula of strongly Drazin inverse under some polynomial conditions,and deduce the theorem for the matrix ring;We study the conditions under which the sum of two elements has strongly Drazin inverse when they have strongly Drazin inverses,and give some examples.In chapter 4,we study strongly Drazin inverse of complex matrices,and determine the sum of two complex matrices with strongly Drazin inverse when they meet the conditions;Apply the theorems to block matrices,we study when complex matrices have strongly Drazin inverse;We than investigate spectral conditions under which complex matrices have strongly Drazin inverse,and apply them to block matrices.Some examples are given to illustrate the effectiveness of the results.In chapter 5,the generalized Cline's formula of strongly Drazin inverse over a ring is further studied and applied to the matrices.Then some examples are given to show the sufficiency and necessity of the condition.Finally,in chapter 6,some related problems are put forward and the research prospect is given.