Researches On System Of Quaternion Matrix Equations And The Generalized Inverse A_T,S²

In this dissertation,by investaging the maximal and minimal ranks of the general solutions to certain systems of quaternion matrix equations,necessary and sufficient conditions for the rank invariance of general solutions to some systems of quaternion matrix equations are given.Then,by the rank method we consider some prosperities of the generalized inverse AT,S(2),such as the reverse order law,the block independence in the generalized inverse AT,S(2),and so on. These results further enrich and develop the quaternion matrix algebra and the theory of generalized inverses.The dissertation is divided into 4 chapters. In Chapter 1,we present the research background and progresses of quaternion,quaternion matrices,quater-nion matrix equations,extremal ranks of matrix expressions,generalized inverse of matrix as well as the work we have done in this dissertation.Some preliminary knowledge used in this dissertation is presented too. In Chapter 2,we estab-lish a new expression of the general solution to the system of matrix equations A1X=C1,XB2=C2,A3XB3=C3,A4XB4=C4.This new expression is not only simple, but also it helps us to overcome the deficencies of the known results. By our new expression,we present the maximal and minimal ranks formulas of the general solution to the system and establish the necessary and sufficent con-ditions for the uniqueness.In Chapter 3,we give the maximal and minimal ranks of the quaternion matrix expression A-B1X1C1-B2X2C2 subject to the consis-tent quaternion matrix equations B3X1C3=A3,B4X2C4=A4;the quaternion matrix expression A-BXDYC subject to the consistent systems of matrix equa-tions A1X=C1,A2X=C2,YB1=C3,YB2=C4,and the quaternion matrix expression A-BA1(i,j,k)DA2(i,j,k)C,respectively.As applications,the necessary and sufficient conditions for three quaternion matrices to be independent in the inner inverse,the consistent conditions for a system of nonlinear matrix equations are given,and some invariant properties with respect to the matrix expressions which contain some generalized inverses are also considered. In Chapter 4,we investigate the rank of the generalized Schur complement with respect to the gen-eralized inverse AT,S(2),then we extend the results to investigate the ranks of some further related matrix expressions such as,the matrix expression involving many independent generalized inverses AT,S(2) and some products of generalized inverses AT,S(2).By our new results,we present some necessary and sufficient conditions for some block matrices to be independent in the generalized inverse AT,S(2);the rela-tionships between the generalized inverse AT,S(2) of the sum of matrices and the sum of the generalized inverse AT,S(2) of each matrix,respectively.some correspond-ing results with respect to the Moore-Penrose inverse,the Weight Moore-Penrose inverse,and the Drazin inverse are also established.