Some Inverse Problem For Matrices With Special Structure And Some Reasearches Of Theory And Algorithm On Constrained Matrix Equations

Matrix inverse problem is an extension of inverse eigenvalue problem. An inverse eigen-value problem concerns the reconstruction of a matrix from prescribed spectral data. Inverse eigenvalue problems arise in a remarkable variety of applications. The list includes but is not limited to control design, geophysics, molecular spectroscopy,particle physics, structure anal-ysis and so on. Procrustes problems involving positive definite and bounded constrained arise in mathematical economics and statistics. The constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set. The study of it has been a hot topic in the field of numerical algebra in recent years. Actually, it is widely used in many fields such as structural design, system identification, structural dynamics, automatics control theory, vibration theory. This dissertation considers two kinds of inverse eigenvlue problems for matri-ces with spectial structure, discusses systematically the Procrustes problems involving positive definite and bounded constrained and some constrained matrix equation problems. The main works and results of this dissertation are as follows.1. The research dicusses the inverse problem and their corresponding optimal approxi-mation of two kinds of new symmetric matrix-(R, S,μ)symmetric and (R, S,α,μ)symmetric, derives necessary and sufficient conditions for the solvability, representation of the general so-lutions, corresponding optimal approximation solutions, studies quantitatively the perturbation analysis for the approximation problem and obtains the specific representation of the perturba-tion bound.2. Based on the Dykstra's alternating projection algorithm, the research studies the Pro-crustes problems involving positive definite and bounded constrained systematically. Some numerical results confirm the efficiency of the presented algorithm. These problems cannot be sloved by conventional matrix decomposition techniquel and CG-type iteration method, for the difficulty is due to the fact that the condition of bounded constrained cannot be expressed by specific representation.3. Based on the alternating projection algorithm, the research constructs iterative al-gorithms to slove the consistent marix equations AX=B, AXB=C, AXAT=B, AX+BY=C and their corresponding optimal approximation over linear subspaces or closed convex set (cone). Many numerical examples show that when the system dimension is large, these algorithms have obvious advantages than some traditional iterative algorithms, such as CG, CGLS algorithm, whether from iteration time or iteration step. When the dimension in-creases exponentially, the iteration step is only required for single-digit growth. These algo-rithms have global convergence, which can obtain the unique solution with minimum norm by choosing the initial matrix as zero matrix, and can get the corresponding optimal approximation solution by choosing the initial matrix as the given initial estimate matrix.4. The research presents an iterative method with short recurrences to slove some real matrix equation problems under central principal submatrices constriant and their correspond-ing optimal approximation problem. For any initial matrix, a least-squares solution of these matrix equations over given matrix set and given central principal submatrices constriant can be determined within finite iteration steps in the absence of round-off errors. The corresponding least-squares solution with minimum norm can be also obtained by choosing a kind of special initial matrices. The research further analyze the theoretical properties of this iterative method and prove that the Frobenius norm of the residual sequence is strictly monotone decreasing. The research also show that the algorithm is stable any case, and give results of numerical experiments that support this claim.This dissertation is supported by the National Natural Science Foundation of China (10571047) and Doctorate Foundation of the Ministry of Education of China (20060532014).This dessertation is typeset by software LATEX2ε.