Solutions To The Matrix Equations With Some Constraints

Matrix equations with constraints have important applications in structural dynamic design,finite element model modification,system parameter identification,nonlinear programming and so on.In this paper,we consider the three kinds of problems of matrix equations with constraints:the constrained solutions of the matrix equation AXB=C,the symmetric solutions of matrix equations K?=M??,?TM?=Ip with subma-trix constraints and its optimal approximation problem,and the solution of the matrix equation AX=ATX? with spectral data constraint and its optimal approximation problem.For the constrained solutions of the matrix equation AXB=C,by imposing some constraints on the expressions of the general solution and the general least-rank solution to the matrix equation and using the Moore-Penrose inverse of matrices,necessary and sufficient conditions for the existences of Hermitian(skew-Hermitian),Re-nonnegative(Re-positive)definite,and Re-nonnegative(Re-positive)definite least-rank solutions to the matrix equation AXB=C are deduced and the explicit representations of the general solutions are given when the solvability conditions are satisfied.For the symmetric solutions of the matrix equations K?=M??,?TM?=Ip with submatrix constraints and its optimal approximation problem,we obtain the necessary and sufficient condition for the existence of symmetric solutions of the matrix equations under submatrix constraints of K₀,M₀ and the explicit representation of the general solution by using QR-decomposition and the generalized singular value decomposition.Then we show the optimal approximation solution for given symmetric matrices Ka and Ma by using the matrix derivatives,the Kronecker product and the vec-function.For the solution of the matrix equation AX=ATX? with spectral data constraint and its optimal approximation problem,by partitioning the eigenvalue matrix ? and using QR-decomposition,the solvable conditions of the problem and the representation of the general solution with spectral data constraint are obtained.Then the optimal approximation solution to the given matrix (?) is obtained by using matrix derivatives.