The Applications Of Partitioned Matrices In Generalized Inverse

The generalized inverse has been a very active research branch of the matrix theorey internationally. It plays an important role in the filed of numerical analysis, differential equations, numerical linear algebra, optimization, control theory, etc. which have wide applications in statistics, matrix theory, network theory, domain decomposition and so on. Moreover, it is a necessary tool in the study of many fields in modern time.In this paper, we study some applications for the block matrices in generalized inverse of matrices theory:Firstly, we presents a full rank factorization of a2x2block matrix. Applying this factorization, we obtain an explicit representation of the group inverse in terms of four individual blocks of the partitioned matrix. We also derive some important coincidence theorems, including in the expressions of the group inverse with Banachiewicz-Schur forms.Secondly, we study the application for the block matrices in the reverse order law for the Moore-Penrose inverse of the product of three bounded linear operators be-tween Hilbert spaces. We first present some equivalent conditions for the existence of the reverse order law (ABC)t=C+B+A+. Moreover, several equivalent statements of R(AA*(ABC))=R(ABC) and R(C*C(ABC)*)=R((ABC)*) are also deducted by the theory of operators.Finally, the representations for the Moore-Penrose inverse and Drazin inverse of the product of orthogonal projection operators P, Q are given by using the expression of the block form of them. Based on the results obtained, the related equalities and properties of the Drazin inverse of the product of P, Q are also given.