Iterative Methods For Procrustes Problems And Two Matrix Perturbation Problems

This thesis mainly studies the following problems:1. Iterative algorithms for the least squares solutions of matrix equation AXB+ CX^TD = E.We propose two iterative algorithms to solve the matrix equation AXB+CX^TD= E. The first algorithm is applied when the matrix equation is consistent. In this case, for any (special) initial matrix X₁, a solution (the minimal Frobenius norm solution) can be obtained within finite iteration steps in the absence of roundoff errors. The second algorithm is applied when the matrix equation is inconsistent. In this case, for the initial matrix X₁ =0, a least squares solution with the minimal Frobenius norm can be obtained.2.The base method and the constraint least squares solutions of matrix equationsWe call some techniques of transforming the constraint problem into the unconstrainted problem as the base method. We propose two techniques and apply them to two problems: the least-squares symmetric solutions of AXB=E with a submatrix constraint and the least-squares centrosymmetric solutions of AXB=E. We characterize the linear mappings from their independent element spaces to the constrained solution sets, study their properities and use these properties to propose a matrix iterative method that can find the minimum or quasi-minimum norm solution based on the classical algorithm LSQR for solving the (unconstrained) LS problem.3.Quaternionic least squares (QLS) problemQuaternionic least squares (QLS) problem is one method of solving overdetermined sets of quaternion linear equations AXB = E that is appropriate when there is error in the matrix E. By means of real representation of a quaternion matrix, we introduce a concept of norm of quaternion matrices, which is different from that in [68, 67], and derive an iterative method for finding the minimum-norm solution of the QLS problem in quaternionic quantum theory.4. Some results on nonlinear equation X^s+A^*X^-qA=IWe analyze the properties on the positive definite solution and deduce some sufficient conditions and necessary conditions. Further, we study the existence conditions and algorithms of some special solution, i.e., the maximal solution, the minimal solution and the quasi-maximal solution. 5. Algebraic properties and perturbation results for the indefinite least squares problem with equality constraintsSeveral equivalent systems without constraints of the indefinite least squares problem with equality constraints (ILSE) are established. We also derive the perturbation results for the ILSE problem and illustrate our results with numerical tests.6. Perturbation analysis for the generalized Schur complement of a positive semi-definite matrixLet P=(?)≥0andS=C-B^HA⁺B be the generalized Schur complementof A≥0 in P. Some perturbation bounds of S are presented which generalizes the result of Stewart[G.W. Stewart, On the perturbation of Schur complement in positive semidefinite matrix, Technical Report, TR-95-38, University of Maryland, 1995], and enrich perturbation theory for the Schur complement. Also error estimate for the smallest perturbation of C, which lowers the rank of P, is discussed.7. The mixed-type reverse-order law of (AB)⁽¹³⁾Some mixed-type reverse-order laws of (AB)⁺ have been proposed and studied by Y. Tian. By applying the PSVD of the matrices A and B, we study mixed-type reverse-order laws of (AB)⁽¹³⁾ and obtain some good results.