Partial Orderings And Generalized Inverse Of Matrices

The generalized inverse of the Fredholm integral operators was given by Fredholm, and the solution of the integral operators equation was obtained. The simpler characterization of generalized inverse was given with four matrices equation by R.Penrose. Since 1950s, many mathematicians have been engaged in studying the' generalized inverse of matrices such as the generalized inverse of matrices on rings, the generalized inverse of morphism, the compution on the generalized inverse of matrices, the application of generalized inverse and so on.The partial ordering of the matrices is facous on the matrix theory, many mathematicians have been engaged in studying the partial ordering of matrix such as kinds of partial ordering and its application.The aim of this paper is to study the generalized inverse of matrices on rings, the generalized inverse of morphism and partial ordering of matrices. The main results are listed in the following:Part l(Chapter2) Some results on the weighted Moore-Penrose of matrices on rings are given, the previous results are extended. The necessary and sufficient conditions for the existence of the -inverse of matrices over the complex fields and its expression are proved with the help of the generalized singular value decomposition. The linear equations APx = b are studied by using the -inverse of matrix A . The expression of the weighted generalized inverse of quaternion matrices are expressed, and the open question on the characterization of the (3,4) and (2,4) inverse are solved.Part 2(Chapter3) The Moore-Penrose inverse and Drazin inverse of morphisms with universal-factorzation in category are studied, its existences are characterized, and the expression of the generalized inverse of morphism are establish. When there is nozero object in category, the generalized inverse of morphisms are studied through the equa-alizer, the necessary and sufficient conditions for generalized inverse is obtained, and the relation between the linear equation and the equalizer is presented in matrix category. A sequence (epic, monic) factorization of morphism is' defined, with the help of the sequence (epic, monic) factorization of morphism, some necessary and sufficient conditions for the Drazin inverse are obtained. We research the generalized inverse of morphisms in preadditive category, give the characterization for the Moore-Penrose and Drazin inverse, and obtain the necessary and sufficient conditions for the existence of core-nipotent for morphism. We defined the generalized Moore-Penrose inverse of morphism, prove it's unique when it is existed, and give some its expression in some cases.Parts 3 (chapter4) The more precision characterization of the star, left-star, right-star and the minus partial orderings are given respectively, and new necessary and sufficient conditions of EP and Hermmite and normal matrices are obtained using the characterization. Certain classes of matrices are indicated for which the star, left-star, right-star and minus partial orderings, or some of them, are equivalent. Some inheritance-type properties of matrices are also given. The relation between some special matrices and its squares are discussed, in the sense of the star partial ordering, the minus partial ordering and Lower partial ordering, the related results of the definite positive matrices are extended, some related results on the general matrices are obtained. The relation between the star and the minus partial ordering of some special matrices is discussed, theprevious related results of the Hermmite matrices are extended. We answer an open question on minus partial ordering posed by J.K.Baksalary and F.Pukesheim, and extend to a general case. Some new properties of left-star and right-star partial ordering are given, the relation between some matrices and its squares are discussed, in the sense of left-star and right-star partial ordering. A new characterization and some properties of the sharp ordering are obtained using the core-nipotent decomposition of the matrices. The D order is...