Representations Of The Drazin Inverses For The Sum Of Two Matrices, Block Matrices And Modified Matrices

Generalized inverse theory of matrix is extremely important and is the new development of the20th century. It has many applications such as in the economic sphere, multivariate statistics, linear programming, network theory, information processing and operations research. The Drazin inverse has numerous applications to singular differential equations and difference equations, to Morkov chains and iterative methods, to cryptography, to numerical analysis, to structured matrices and to perturbation bounds for the relative eigenvalue problems. According to the type of generalized inverse, the content of this thesis can be divided into two parts. The first part is focused on the Drazin inverse, such as the Drazin inverse of the sum of two matrices, the Drazin inverse of some block matrices and the Drazin inverse of modified matrices. In the second part, we consider the W-weighted Drazin inverse of modified matrices. The structure of the thesis is as follows:In Chapter2, we give some explicit formulae for the Drazin inverse of P+Q under some different conditions given in recent articles on the subject and numerical examples are given to demonstrate our results. Using the representations of the Drazin inverse of P+Q, in chapter3, we study the Drazin inverse of block matrix M={ABCD}(where A and D are square matrices) under some assumptions, which can be regarded as the generalizations of some results given in literature. Also numerical examples are given to illustrate our block matrices results.In Chapter4, we present some expressions for the Drazin inverse of a modified matrix M=A-CDDB in terms of the Drazin inverse of the matrix A and the generalized Schur complement Z=D-BADC under some conditions, which extends some existing results in the literature. Further, we study some new results for the Drazin inverse of the modified matrix M=A-CDDB, when the generalized Schur complement Z=0under some assumptions. Numerical examples are given to demonstrate our results.In Chapter5, we establish some representations for the W-weighted Drazin inverse of the modified matrix M=A-CWDd WWB in terms of the W-weighted Drazin inverse of the matrix A and the generalized Schur complement Z=D-BWAd,w WC under some assumptions. Also we obtain some expressions for the W-weighted Drazin inverse of the modified matrix M A CBunder some conditions. Finally, we give twonumerical examples to illustrate our results.