Generalized Inverse And Weighted Generalized Inverse Of Matrices Over Rings

The theory and methods of generalized inverse of matrice are important basic tools in many mathematical desciplines,and have extensive applications in economics,statistics,surveying,optimization techniques,information processing,operations research and so on.Generalized inverse matrices are indispensable studying tools in the least square problems,the linear or nonlinear problems,the nonconstrained or constrained linear programming problems,control and identification of system problems and so on.On the other hand,the study of algebra structure of associative ring is prevalent,and generalized inverse matrice over rings is an important tool in revealing algebra structure of associative ring.In recent years,with the in-depth study,the study of generalized inverse of matrices over field,division rings,PID,Artin rings have been developing.We give some spreaded conclusions and summaries on the base of theroys in essay[6],[7],[12],etc.In character three,the Moore-Penrose inverses of matrices over a ring R with an involution and unitary are discussed.Some necessary and sufficient conditions for the existence of the Moore-Penrose inverse of matrices over a ring R are given.In particular,it is obtained that a necessary and sufficient condition for the existence of the Moore-Penrose inverse of a matrix A which have the form of A=GDH and D²=D=D^* over a ring R.But under normal circumstances,D may be not idempotent,so we can discuss the necessary and sufficient conditions for the existence of generalized inverse of matrices over ring.Also we can give some corresponding expressions,and the necessary and sufficient conditions of the {1,3}-inverse and {1,4}-inverse.If matrices over rings don't have the special form,theorem 3.2.2 gives the necessary and sufficient conditions of the general matrix's generalized inverse:(1)There exists X s.t.AXA= A;(2)There exists Z₀ s.t.A=AA^*Z₀;(3)There exists Y₀ s.t. A=Y₀A^*A.In character four,we mainly discuss the weighted generalized inverse with respect to M and N,and gives the necessary and sufficient conditions for the existence of the weighted generalized inverse with respect to M and N with some decomposable matrices.In this condition,theorem 4.1.6 gives the necessary and sufficient condition:(GD)^*M^*GD+I-D^-D,DHN^-1(DH)^*+I-DD^-are both symmetrical and invertible.We also give the corresponding expressions.When the MP-inverse of D exists,it is clearly that the {1}-inverse of D exists,so the conclusion's conditions we need is weaker than the conditions in essay[9],and then the conclusions is a generalization of[9].At the same time,we can also give the necessary and sufficient conditions of the {1,3M}-inverse and {1,4N}-inverse of matrices satisfying the same conditions.At last,we extend the necessary and sufficient condition for the existence of the MP-inverse of matrices over a ring with an involution and unitary to the weighted MP-inverse.