Exchange Of The Generalized Inverse Of Matrix In The Ring With A Partial Order

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 1(Chapter2) The relationship arediscussed between a matrix and its bordered matrix over commutativering in terms of subdeterminants. Necessary and sufficient conditions are given, for the existence of the Moore-Penrose inverseand the Drazin inverse of a matrix over commutative ring, by itsbordered matrices. Representations of the Group inverse and theDrazin inverse are given in terms of subdeterminants, for the existence of theГ-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 theГ-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.Parts 2 (chapter3) 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 normal matrices are obtained using the characterization. The relation between normal 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. A new characterization and some properties of the sharp ordering are obtained using the core-nipotent decomposition of the matrices. The D order is defined and some properties are discussed. Some mistakes in [120] are corrected. We ponit out a mistaken on the chracterization of relation between the minus and the star partial ordering by J. K. Baksalary and Jan. Hauke.Part 3(Chapter4) 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.