The Least-squares Constrained Solutions Of The Matrix Equation AX+YB=E And Its Optimal Approximation

The constrained matrix equations, least-squares problems and the related optimal approximation problems have been a lot of applications in many fields, including particle physics and geology, the inverse problems of control theory, the inverse problems of vibration theory, the approximation problem over finite elements and multi-dimensional space etc.We have studied the least-squares constrained solutions of the matrix equation AX+ YB=E and its related optimal approximation, which can be described asProblemⅠ. Given matrices A,B,E∈Rm×n,and S1 (?)Rn×n,S2(?)Rm×m, find X∈S1,Y∈S2 such that‖AX+YB-E‖=min where S1,S2 are the sets of real matrices, symmetric matrices, skew-symmetric matrices, centro-symmetric matrices and skew-centro-symmetric matrices of suitable size respec-tively.ProblemⅡ. GivenA, B, E∈Rm×n,X∈Rn×n,Y∈Rm×m;find[XY]∈SE,such that where SE is the solution set of problemⅠ,‖·‖is the Frobenius norm.First the paper has presented the concept of matrix pair, the inner product of ma-trix pairs and the subordinated norm of matrix pair. Secondly, applying the projection of finite dimension subspace, we obtain the normal equations of‖AX+YB-E‖= min under the constrains of real matrix, symmetric and skew-symmetric matrix, centro-symmetric matrix and skew-centro-symmetric matrix respectively. Furthermore, by us-ing the idea of conjugate gradient iteration method and the structure characteristic of matrices, we construct iterative methods to compute the real solutions, symmetric and skew-symmetric solutions, centro-symmetric and skew-centro-symmetric solutions of‖AX+YB-E‖=min respectively. We prove that this methods will get the solutions within finite iterative steps without roundoff errors. And by choosing the special initial matrices, this methods can also compute the corresponding optimal approximation to given matrix pair. Some numerical examples show that this methods work very well.