The SMW Formula And Displacement Structure Of Some Generalized Inverse Of Matrices

Various important and interesting generalized inverses appear since the discovery of the relationships between the least-squares properties of generalized inverses and the solutions of linear systems,such as Moore-Penrose inverse,Drazin inverse,group inverse and core inverse.The theory and computation methods of generalized inverses have been developing rapidly in the past few decades,different kinds of representations and numerical computations of generalized inverses are emerging in an endless stream.Moreover,the applications of various generalized inverses can often be seen in scientific calculation,statistics,image processing,engineering,economics.In this paper,we study some related theoretical and computational problems of the core inverse,DMP inverse,mainly including(1)By using the inclusion relations between the range and the kernel,some conditions under which the Sherman-Morrison-Woodbury formula for core inverse holds are given,and the classical Sherman-Morrison-Woodbury formula is extended.The nonnegativity of the core inverse of perturbed matrix is discussed by using the Sherman-Morrison-Woodbury formula of core inverse.(2)The displacement structure of DMP is studied.Estimations for the Sylvester displacement rank of DMP inverse are presented under some restrictions.The generalized displacement is also discussed.The general results are applied to the core inverse.