Research On Reverse Order Law For Generalized Inverse And Drazin Inverse

The generalized inverse is a very important discovery in the matrix theory in the last century, especially, the research in theories and computational methods of generalized inverse of matrices have been made fast development from1950s, and it has extensive applications in mathematical statistics、mathematical programming、 econometrics、numerical analysis、game theory、control theory、network theory and other disciplines. With the development of generalized inverse, the Drazin inverse and the other type generalized inverses were found. The Drazin inverse was shown to be very useful in various applications in los of fields, for example, applications in singular differential or difference equations, operator theory, Markov chains,cryptography, iterative method, and so on. 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 Moore-Penrose inverse and the generalized inverse associated to the Moore-Penrose inverse, including reverse order law for generalized inverse、common least-squares solution、partial orderings, etc.; in the second part, we consider the Drazin inverse, such as the Drazin inverse of the sum of matrices、the Drazin inverse of a modified matrix and the Drazin inverse of block matrix. The structure of the thesis is as follows.In chapter2, we study the ordinary reverse order law for{1,2,3}-inverse and{1,2,4}-inverse of the multiple-matrix products, based on a rank equality related to {2}-inverse, we show that An{1,2,i}…A2{1,2,i}A1{1,2,i}∈(A1A2…An){1,2,i} is equivalent to An{1,2,i>…{1,2,i}A1{1,2,1}=(A1A2…An){1,2,i} for i=3,4, which completely solve the bequeath problems about these two reverse order law.In chapter3, we consider the mixed-type reverse-order laws for{1,2,i}-inverse (i=3,4) of two matrices products by using the extremal ranks of generalized Schur complements, and establish the sufficient and necessary conditions for the following relations to holdIn chapter4, firstly, we derive the conditions for the existence a common least-squares solution to the pair of matrix equations A1XB1=C1and A2XB2=C2, and the explicit representation of the general common least-squares solution is obtained. Furthermore, we use this result to determine the condition for the existence of a Hermitian least-squares solution to the matrix equation AXB=C, and the expression of the general Hermitian least-squares solution is also given. Finally, we consider the existence of Hermitian{1, i}-inverses of A, i=3,4, and the representations of the Hermitian generalized inverses are presented.In chapter5, we give a representation of the Drazin inverse of P+Q under the conditions PQP=0, PQ2=0(or QPQ=0, P2Q=0). Additionally, suppose A is an idempotent matrix, the Drazin inverse of the modified matrix A-CB is also considered. Using the a representation of the Drazin inverse of P+Q, in chapter6, we study the Drazin inverse of a2×2block matrix with some weaker conditions, which improved some known results; moreover, the group inverses and the Drazin inverses of a class of2×2anti-triangular block matrices are investigated.In chapter7, we mainly study the partial orderings of rowwise partitioned matrix and columnwise partitioned matrix, i.e., the relation between A(?)C, B(?)D and Specially, the star ordering of a sum of two matrices and the minus ordering of matrix product are also studied, and the conditions for the two partial orderings to hold are given.