Research And Application Of The Theory Of Generalized Inverse Matrix

The generalized inverse matrix has a distinctive advantage in solving the matrix equation, what’s more, its applications in numerical analysis, cryptography, astronomy,economic management and other fields are gradually being recognized. Currently, the generalized inverse matrix has become an important branch of matrix theory. On the other hand, the study of algebraic structure of associative ring is a focus of theoretical study and generalized inverse over rings is an important tool in revealing algebraic structure of associative ring.The Drazin inverse or group inverse of block matrix is an important part of the theory of generalized inverse matrix. Until now seeking the representations of the Drazin inverse or group inverse for arbitrary block matrix is still an unsolved problem. Giving some appropriate conditions or weakening the original conditions to obtain the representation of Drazin inverse or group inverse of block matrix is a meaningful job.In this paper, a new kind of weighted Moore-Penrose inverses of matrices over rings with involution*, the Drazin inverse and group inverse of block matrix under certain conditions are studied. Finally, some applications of the generalized inverse matrix are given, mainly used in the calculation of resistance distance. The main results are as follows:(1) Using the ring theoretic methods, a sufficient and necessary condition and some related equivalent conditions for the existence of this kind of weighted generalized inverse are given. The expression is also given.(2) The Drazin inverse of block matrix is discussed and the expressions under the conditions of BDπC=0,DDπC=0,BDDDC=0,BDDDD=0 and AB=CB=0,DπCB=0 and AπBC=0,AπBD=0, CAAπ=0,CAπB=0,AADB=0 and CAD=0,DπCB=0,CBDD=0,Aπ(A-BDDC)B=0 is given.(3) The group inverse of block matrix is discussed and the expression of group inverse of trim matrix is given when B is a row full rank matrix.(4) Some applications of generalized inverse are discussed. combining the theory of generalized inverse matrix with the knowledge of graph theory, we give the distance of resistance between any two points through the generalized inverse of Laplace matrix of the graph. Furthermore,the distance of resistance between any two points in any weighted graph is given by Matlab.