Study Of Some Problems On Generalized Inverse Matrices

The theory and methods of generalized inverse matrices are important basic tools in all mathematical disciplines, and have extensive applications in economics, statistics, surveying, optimization techniques, information processing, automatic control, engineering techniques, operations research and so on. Generalized inverse matrices are indispensable studying tools in the least-square problems, the rectangular or ill-linear problems, the nonlinear problems, the non-constrained or constrained linear programming problems, control and identification of system problems, electronic net problems and so on. On the other hand, the study of algebra structure of associative ring is prevalent, and generalized inverse matrices over a ring is an important tool in revealing algebra structure of associative ring. In recent years, with the study of theory and computation of generalized inverse matrices, there are a series of open problems in the fields of generalized inverse matrices.In this paper, by applying the methods of representation of generalized inverse matrices and the methods of matrix decomposition, we will study and settle following three problems on generalized inverse matrices:1.The rank equalities problems related to the generalized inverse AT,S(2)The rank equalities problems related to the generalized inverse matrices are important subjects. It is very important for characterizing some equalities related to the generalized inverse matrices. It is well-known that M-P inverse A+, the Drazin inverse AD, the Weighted M-P inverse AM,N,+, the group inverse Ag, the Bott-Duffin inverse A(L)(-1) and the generalized Bott-Duffin inverse A(L)(+) are all generalized inverses AT,S(2). By applying the group inverse representation of generalized inverse AT,S(2), we study the rank equalities problems related to the generalized inverse matrices, we obtained some rank equalities related to the generalized inverse AT,S(2) of a matrix A, to the generalized inverse AT,S,(2),BT1,S1(2) of two matrices A and B and some rank equalities for submatrices in generalized inversesMT,S(2) of a partitioned matrix M. As corollaries, we obtained some characterizations related to M-P inverse , the Drazin inverse , the Weighted M-P inverse, the group inverse , the Bott-Duffin inverse and the generalized Bott-Duffin inverse.2. The problems of the block independence in generalized inverse of a partitioned matrixThe problems of the block independence in generalized inverse matrices of partitioned matrices have been studies by many authors, but some open problems still are not solved. By applying the multiple quotient singular value decomposition QQQQQ-SVD, We prove a conjecture appeared in SIAM J.Matrix Anal.Appl in 1998.3.The problems of generalized inverse matrices over a division ring The generalized inverse matrices over a division ring are important for studyingalgebra structure of a ring and generalized inverse of quaternion matrices. By applying the methods of matrix decomposition, we obtained general theory of generalized inversesTT,S(2) over a division ring, and studied the reverse order laws for generalized inverses of matrices, we also studied the structure of generalized inverse of a bordered matrix. As corollaries, the generalized inverse matrices over quaternion skew field can be obtained.