The Solutions Of The Matrix Equations On A Subspace

Matrix equations are widely used in parameter identification and structural dynamic design and so on.In this paper,we consider the solutions of three kinds of matrix equations on sub space:the solution of the matrix equation AXA^（?）=B and its associated optimal approximation problem;the symmetric solutions of the matrix equation AXB=D and the corresponding optimal approximation problem;the common solutions of the matrix equations AX=B,XC=D and the corresponding optimal approximation problem.For solving the solution of the matrix equation AXAT=B and its associated optimal approximation problem,we obtain the solvability conditions and the expression of the general solution for the matrix equation AXA^（?）=B on the subspaceΩ={z∈Rⁿ|Gz=0,G∈R^k×n} in the light of the generalized singular value decomposition of a matrix pair,and achieve the optimal approximation solution to the given matrix （?） by means of the Hadamard product of matrices.For solving the symmetric solution of the matrix equation AXB=D and the corresponding optimal approximation problem,we deduce the solvability conditions and the general solution expression of the matrix equation AXB=D on the subspaceΩ={z∈Rⁿ|Gz=0,G∈R^k×n} by applying the Moore-Penrose inverse of matrices,and show the optimal approximation solution to the given matrix （?）₂ by using the Kronecker product of matrices and straightening function.For solving the common solution of the matrix equations AX=B,XC=D and the corresponding optimal approximation problem,we discuss the necessary and sufficient conditions for the existence of the common solution to the matrix equations AX=B,XC=D on the subspace Ω={z∈Rⁿ|Gz=0,G∈R^k×n} by means of the generalized inverse of matrices,and give the explicit representation of the general solution when the solvability conditions hold.Moreover,we elucidate the optimal approximation solution to the given matrix （?）₃ on the basis of the matrix derivative and the Kronecker product.