Solving Several Kinds Of Matrix Equations By Numerical Methods

Matrix equations are widely applied in state covariance assignment,time series analysis,structural dynamics model modification,system parameter identification and so on.In this paper,we consider the four kinds of problems:the common solution to the matrix equations AXB=D,GXH=C the least-squares solutions of the matrix equationsAX₁+X₂ A^H+BY₁ C+C ^HY₂ B^H=E,A^HX+X^HA+B^HYC+C^HY^HB=D the column unitary solutions of the matrix equation AXB=D and the system of matrix equations AX=C,XB=D and their optimal approximation problems the Re-nonnegative definite and Re-positive definite solutions to the matrix equations AX=C,XB=D.For the common solution of the matrix equations AXB=D,GXH=C,by imposing the Moore-Penrose inverse of matrices,necessary and sufficient conditions for the existence of the common solution to the matrix equations AXB=D,GXH=C are deduced and the explicit representations of the general solution are given when the solvability conditions are satisfied.For the least-squares solutions of the matrix equationsAX₁+X₂ A^H+BY₁ C+C ^HY₂ B^H=E,A^HX+X^HA+B^HYC+C^HY^HB=D,by using the singular value decom-positions of the matrices A and canonical correlation decompositions of some matrix pairs,the representations of the least-squares solutions to above two kinds of matrix equations are deduced,respectively.For the column unitary solutions of the matrix equation AXB=D and the system of matrix equations AX=C,XB=D and their optimal approximation problems,by utilizing the spectral decompositions and singular value decompositions,necessary and sufficient conditions for the existences of the column unitary solutions to above matrix equations are deduced and the explicit representations of the general solutions are given when the solvability conditions are satisfied.Then the optimal approximation solutions to the given matricesXare showed by using the singular value decompositions.For the Re-nonnegative definite and Re-positive definite solutions of the matrix equations AX=C,XB=D,by using the Moore-Penrose inverse and generalized singular value decomposition of matrices,necessary and sufficient conditions for the existences of Re-nonnegative definite and Re-positive definite solutions to the matrix equations AX=C,XB=D are deduced and the explicit representations of the general solutions are given when the solvability conditions are satisfied.