Combined With The Generalized Inverse Of A Ring <sub> T, S </ Sub> ~ (2) Theory And Computing

The concept of generalized inverses of matrices was introduced by E.H. Moore in 1920. From then on, the theory of generalized inverses develops and fields of application expand incessantly, including statistics, cybernetics, dynamical system, solutions of non-linear equation, optimization, graph theory and combinatorics.In 1974, Ben-Israel and Greville introduced the concept of the generalized inverse A_T,S² of a matrix A (the {2} inverse with prescribed range and null space). Since Such the known generalized inverse as the Moore-Penrose inverse, the Drazin inverse, the group inverse and so on, is the generalized inverse A_T,S²,we can obtain the properties of every concrete generalized inverse, find the commonness of every known generalized inverse, unify the computing formulas of every concrete generalized inverse and otherwise by studying the generalized inverse A_T,S².In the thesis, we study mainly the existence of the generalized inverse A_T,S² of a matrix A and explicit expressions for A_T,S² over some associative rings, minors of A_T,S² over an integral ring, and some applications. The thesis is divided into seven chapters.In Chapter 1, we give some required preparative knowledge, including basic concepts and general symbols. In this chapter, we also introduce the concept of the weighted Moore-Penrose inverse over an associative ring, which extends that defined by K.M. Prasad and R.B. Bapat over an integral domain. In addition, we also prove that if the weighted Moore-Penrose inverse exists, then it is unique, and discuss the basic properties of the weighted Moore-Penrose inverse.In Chapter 2, we introduce the concept of the generalized inverse A_T,S² of a matrix A over an associative ring, study the properties of A_T,S², and verify that weighted Moore-Penrose inverses, Moore-Penrose inverses, Drazin inverses and group inverses, if exist, are all A_T,S² for an arbitrary matrix A over an associative ring.In the chapter, we give also the necessary and sufficient condition for the existence of the generalized inverse A_T,S^1,2.In the rest of this chapter, we discuss A_T,S^1,2 of a von Neumann regular matrix A over an associative ring. We give the sufficient conditions for the existence of A_T,S^1,2 of a von Neumann regular matrix A, explicit expressions for A_T,S^1,2 which reduce to group inverses or {1} inverses. We also show that A₍(R(G),N(G))^1,2 of a von Neumann regular matrix A is the same as the known generalized inverse of A for an appropriate matrix G under some conditions over an associative ring. In addition, we also deduce the equivalent conditions for the existence of the weighted Moore-Penrose inverse over an associative ring. In Chapter 3, we study the generalized inverse A_{T, S}⁽²⁾ of a matrix A over an integral domain. Firstly, we give some necessary and sufficient conditions for the existence of A_{T, S}⁽²⁾, an explicit expression for the elements of A_{T, S}⁽²⁾ and explicit expressions for A_{T, S}⁽²⁾. Secondly, we discuss the relation between the known generalized inverses and A^<sub>T, S⁽²⁾. Thirdly, we give the relation between some rank equation and the existence of the generalized inverse A_{T, S}⁽²⁾. Finally, we give a example of evaluating the elements of A_{T, S}⁽²⁾ without calculating fully the generalized inverse A_{T, S}⁽²⁾.In Chapter 4, we discuss the minor of the generalized inverse A_{T, S}⁽²⁾ of a matrix A over integral domains and establish an explicit expression formula for minors of the generalized inverse A_{T, S}⁽²⁾. It is convenient to study submatrices of the generalized inverse A_{T, S}⁽²⁾ over integral domains.In Chapter 5, by using the determinant rank of matrices, we study conditions of the reserve order law for the generalized inverse A_{T, S}⁽²⁾ of the product of three matrices over integral domains.In Chapter 6, we show further the properties of the generalized inverse A_{T, S}⁽²⁾ of the matrix A over skew fields. We give a necessary and sufficient condition for the existence of the generalized inverse A_{T, S}⁽²⁾ explicit expressions for A_{T, S}⁽²⁾, which reduce to {1} inverses, and verify that group inverses, Drazin inverses andρMoore-Penrose inverses are identical with the generalized inverse A_{R(G), N(G)}⁽²⁾ for an appropriate matrix G, respectively.In this chapter, we also discuss the reverse order law for the generalized inverse A_{T, S}⁽²⁾ over a skew field.In Chapter 7, we study the calculation of the generalized inverse A(x)_{T, S}⁽²⁾ of a polynomial ma- trix A(x) and propose an algorithm based on the discrete Fourier transform to compute A(x)_{T, S}⁽²⁾. The algorithm has its origin in algorithms proposed by Karampetakis and Vologiannid for com- puting the Moore-Penrose inverse and the Drazin inverse in 2003. In order to deduce the algorithm for computing A(x)_{T, S}⁽²⁾, we first give a new proof of the theorem which is based by the finite algo- rithms for the generalized inverse A_{T, S}⁽²⁾ of a constant matrix A. And thus, analogously, we obtain the theorem based by the finite algorithms for the generalized inverse A(x)_{T, S}⁽²⁾ of a polynomial matrix A(x).In the end of this chapter, we list the program code of the algorithm, written in the Mathe- matica programming language.