The ～*Congruence Class Of The Solutions Of Matrix Equations And Generalized Inverses Of A Normal Matrix

The theory and methods of generalized inverse are important basic tools in all mathematical disciplines, and have extensive applications in economics, statistics, surveying, optimization techniques, information processing, automatic control, engineering techniques, operations research and so on. Especially, generalized inverse matrices are indispensable studying tools in least square problems, the rectangular or ill-linear problems, the nonlinear problems, the non-constrained or constrained linear programming problems, control and identification of system problems, electronic net problems and so on. Linear matrix equations frequently arise in many areas of scientific computing and engineering applications, for example, the computations of inverse eigenvalue problems, the finite element model updating in structural dynamics, the constructions of sparse approximate inverse preconditioners for Krylov subspace iteration methods and etc.In this paper, the *congruence class of a least square solution for the following matrix equations AX = B, A*XA = D, AXB = D, (AX XB) = (E F) is presented. Also, we derive a necessary and sufficient conditions for the existence of a least square solution and present a general form of such solutions using the Singular Value Decomposition (SVD) and Generalized Singular Value Decomposition(GSVD). The normal matrix is a very important class in matrix analysis. In this paper, the expressions for generalized inverses of a normal matrix are discussed by its Schur decomposition.