The Theory And Algorithm For Solving A Class Of Matrix Nearness Problem

The problems of solving linear matrix equations and the corresponding least-squares problems have been a hot topic in the field of numerical algebra in recent years, and have been widely applied in many fields such as structural design, system identification, structural dynamics, automatics control theory, vibration theory. The matrix nearness problem, occurring in experimental design and the finite element model updating problem, consists of finding a nearest member X in a class of special matrix set to a given matrix X^*, where distance is measured in a matrix norm. A kind of matrix nearness problem arising in the structure dynamics model updating problem is systematically studied in this dissertation:Problem I Given matrix X*∈R^n×m, find X|^∈S such thatwhere ||·|| stands for the Frobenius norm, and S denotes the least-squares solution set of the following inconsistent matrix equationsandover the common matrix set or symmetric matrix set, respectively.In this dissertation, the direct method associated with several matrix decompositions and the iterative method with shot recurrences are utilized to find the solutions of Problem I, respectively, and the main works and results are as follows:1. Based on the projection theorem in the finite-dimensional inner product space, by making use of the generalized singular value decomposition (GSVD) and the canonical correlation decomposition (CCD) of matrix pair simultaneously, we transform the least-squares problems of the abovementioned inconsistent matrix equations into the problems of solving the consistent matrix equations over given matrix set, and obtain the general expressions of the corresponding least-squares solutions. Combing these expressions with the orthogonal invariance of the Frobenius norm, we overcome the key difficult for the matrix unitary approximation successfully, and obtain the expression of the solution of Problem I. Furthermore, the numerical algorithms and examples to solve Problem I are given. 2. An iterative method with short recurrences is presented to solve the matrix nearness problem associated with the abovementioned inconsistent matrix equations. For any initial matrix, a least-squares solution of these matrix equations over given matrix set 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. Moreover, Problem I can be transformed equivalently into a problem that finding the minimum norm least-squares solution of a new inconsistent matrix equation.3. We further analyze the theoretical properties of this iterative method. We construct a kind of special matrix function, which is used to characterize the minimization property of this iterative method, and prove that the approximate solution, generated by this iterative method, minimizes this kind of matrix function over a special affine subspace, which means that the Frobenius norm of the residual sequence is strictly monotone decreasing. Analogous with the classical conjugate gradient method, we derive a rough error bound by means of the minimization property of this iterative method. Finally, we give several numerical examples to verify the obtained theoretical results.The conventional matrix decomposition methods have been used to find the least-squares solutions of the abovementioned inconsistent matrix equations over given matrix set in many references, and the general expressions of these solutions were obtained. But it is difficult to determine the solution of Problem I by utilizing these expressions due to the fact that the orthogonal invariance of Frobenius norm does no(?) hold for the general nonsingular matrices. In this dissertation, two matrix decomposition methods are applied simultaneously to overcome this difficult skillfully, and the expression of the solution of Problem I is also obtained. We also adopt the iterative method to study the matrix nearness problem associated with these inconsistent matrix equations systematically in this dissertation, and characterize the minimization property of this iterative method by making use of the matrix function in subspace for the first time, which is the important improvement for the achievements of references.