The Research Of Perturbation Of The Drazin Inverse Of Matrices

The Drazin inverse of matrices is a very important part in the theory of generalizedinverses. It is applied widely, for example, the solution of singular linear systems, thetheory of finite Markov chain, control theory, numerical analysis and so on. The Drazininverse has obviously been the necessary tool in many research fields. In recent years,many researchers were devoted to studying the perturbation of the Drazin inverse andthe expressions of Drazin inverse of sum of matrices. But all of these were given underthe restrict conditions. In this paper, we continue to study both of them under somenew conditions. The paper is organized as follows:In chapter1, we introduce the background, required general symbols, definitionsand basic lemmas. Also, the main results of the paper are presented.In chapter2, we discuss the expressions of （A+E）Dwhen the blocks of matricesA，E satisfy various conditions. Further, we give the perturbation upper bounds withrespect to （A+E）D. Also, we compare the perturbation upper bounds obtained in thischapter with some important ones in the references by using an example.In chapter3, we deduce the explicit expressions for （P+Q）^Dand （P Q）^Dof twomatrices P and Q under the conditions P²Q=P QP and Q²P=QP Q. Notice thatthese two conditions are weaker than P Q=QP in [31], and the result is generalized.Also, we give the upper bound of||（P+Q）^D P^D||₂.