Least-squares Solution For The Inverse Problem Of Several Matrices

Least-squares solution for the inverse problem of real matrices,symmetric matrices and bisymmetric matrices are studied in this thesis. The thesis includes the following contents.Firstly, least-squares solution for the inverse problem of real matrices with a submatrix constraint is proposed. Least-squares problem of the real matrices is discussed and the expression of general solution is given. The best approximation to a given matrix is considered. The existence and uniqueness of the optimal approximation are proved. A numerical method for finding the optimal approximation is presented. These results are applied to solve a class of inverse eigenvalue problems for real matrices with a submatrix constraint and the expression of general solution and best approximation are also given.Secondly, least-squares solution for the inverse problem of real symmetric matrices with a submatrix constraint is proposed. Least-squares problem of the real symmetric matrices is discussed and the expression of general solution is given. The best approximation to a given matrix is considered. The existence and uniqueness of the optimal approximation are proved and the optimal approximate solution to a given matrix in the solution set is also given.Finally, least-squares solution for the inverse problem of bisymmetric matrices with a submatrix constraint is proposed. Using the generalized singular value decomposition(GSVD), the expression of general solution is given. The best approximation to a given matrix is considered. The existence and uniqueness of the optimal approximation are proved. The optimal approximate solution to a given matrix in the solution set is also given.