Research For The Properties,Computaton Of Generalized Inverse And Some Linear Systems

The thoery of generalized inverses and its computation,the solution of linear systems developed in 1930s.There are a lot of results about them.The generalized inverses of a matrix have wide applications in many areas,such as differential and integral equtations, operator theory,control theory,optimal theory,Markov chains,and etc.And linear systems also have many application in many eares,such as experimental design,control and commutation theory,statistics analysis,science and technology engineering and so on. Now,both of them are important tools in science computation.In this paper,A further researches are considered for the representation,computation and perturbation of generalized inverses of a matrix and some new results obtained.Different methodes to solve some linear systemes are also studied.Two main kinds of problems have been studies and resolved as follows:1.The representation,computation and algebra perturbation of the generalized inversesThis part consists of two chapters.In th second chapter,some method to compute the generalized inverses are introducted,in the first section,a novel representation for the generalized inverse A_T,S²based on Gausse elimination is given,and corresponding algrithm and arithmetic operation are also summarized.In the second section,we first introduce a full-rank representation of the generalized inverse A_T,S²of a given complex matrix A,then two affine combination expression of the generalized inverse A_T,S²are also studied,then we give some application of these representations.In the third section,an weighted conjugdate gradient method is constructed,used this method,a finited iterative formulae is presented for weighted M-P inverse and M-P inverse.In the fourth section we further the properties of line and column of a partitioned matrix,and give some representations of M-P inverse,group inverse and Drazin of the partitioned matrix.Based on this,we get three kind quotient identities of generalized Schur complement about the partitioned matrix.In the third chapters,algebra perturbation and modifaction theory are studies.In the first section,we first study the range and null space of a rectangular matrices,the expressions of the algebraic perturbation are given.These results develop the theories in some papers.In second section,the algebra perturbation and analysis perturbation of weighted group inverse for rectangular matrices are studied.In the third section,algebraic perturbation theory for theα-βgeneralized inverse is disscussed.In the fourth section, we use the thoery of modification of rank-1 of M-P inverse,get the representattion of modification of rank-1 for Boot-Duffin inverse and generalized Boot-Duffin inverse.2.The solution of some linear systemsThis part also consists of two chapters.In fourth chapter,the problem of optimal approximation and solution for some matrix equations are researched.In the first section, an efficient iterative method is presented to solve a pair of linear matrix equations (AXB,CXD)=(E,F).In second section,we use the theory of generalized inverse and projection method to study all kind solution of matrix equation AX=B,XD=E, which is different to the decomposition of matrix.In the third section,we give a polynomial method to solve the matrix equation AXB+CXD=E,which develops the results in some papers.In last section of this chapter,The theory of a modifications of steepest descent algorithm and minimal residual iterative are studied.In the fifth chapter,we main study the indefinite least square problem(ILS),we first define a new generalized invers—generalized weighted M-P inverse(also named indefinite least square generalized inverse) and study some properies of it.Then we use this generalized inverse to draw an explicit solution of ILS.