Generalized Inverses Of Matrices

In this paper, a new kind of weighted Moore-Penrose inverses of matrices over rings with involution ?,the weighted generalized inverses of fuzzy matrices, the Cline inverse of fuzzy matrix, the Cline inverse of incline matrix and a new kind of weighted Moore-Penrose inverse of incline matrix are studied. The main results are listed as follows:(1) An answer to an open problem put forward by Robert E.Hartwig in [90] is given. A new kind of weighted Moore-Penrose inverses of matrices over rings is defined and some properties are obtained by using ring theory methods here. When the powers are both unit matrices, it is the common Moore-Penrose inverse. When R is complex field, it is the same as theΓ-inverse of a matrix in [84].(2) The weighted Moore-Penrose inverse of fuzzy matrix is defined. If the weighted Moore-Penrose inverse of fuzzy matrix A exists, then it is unique. Some necessary and sufficient conditions for the existence and the expressions of AM + , N are given. Then the reverse order law of the weighted Moore-Penrose inverse of fuzzy matrix is introduced. Last, an example is given to illustrate that while the weighted Moore-Penrose inverse of fuzzy matrix A exists, the Moore-Penrose inverse of fuzzy matrix A does not necessarily exist.(3) We investigate the inverse of fuzzy matrix. First, we introduce the Cline inverse and the Drazin inverse of the fuzzy matrix, with the properties of fuzzy matrix we prove that any fuzzy matrix has the Drazin inverse, thus we get any fuzzy matrix has the Cline inverse, and it is unique. Moreover, if A+ exists, then AC equals to A+ . Last, we give some properties of AC .(4) We investigate the inverse of incline matrix. First, we introduce the Cline inverse and the Drazin inverse of incline matrix, with the properties of incline matrix we prove that any incline matrix has the Drazin inverse, thus we get any incline matrix has the Cline inverse, and it is unique. Moreover, if A+ exists, then AC equals to A+ . Last, we give some properties of AC .(5) A new kind of weighted Moore-Penrose inverses of Incline matrices is defined and some properties are obtained.