The Symmetric Or Antisymmetric Solution Of AX=B In D Inner Space And The Optimal Approximation

The problem of solving a linear matrix equation is an important study of the nu-merical linear algebra. It has been widely applied in biology, electricity, molecular spec-troscopy, vibration theory, finite elements, structural design, solid mechanics, parameteridentification, automatic control theory, linear optimal control and so on.This master thesis is mainly concerned with the problem how to get the least squaressolutions of the matrix equation AX = B and its optimal approximation for D symmetricmatrices and D antisymmetric matrices in D inner space.Problem I. Given X, B∈R_D^n×m, find A∈S such thatAX = B.Problem II. Given X, B∈R_D^n×m, find A∈S such thatProblem III. Given A?∈R_D^n×n, find A∈SA such thatwhere S is the set of D symmetric matrices or the set of D antisymmetric matrices, SAis the solution set of problem I or the solution set of problem II,·F is the Frobeniusnorm,·D is the D norm.The paper has four parts.In the first chapter, the background, significance and progress situation for inverseeigenvalues problems, linear restriction problems, least squares problems and the relatedoptimal approximation problems are presented. And the main work of this paper is alsosimply introduced.In the second chapter, the properties of the D symmetric matrices and the D anti-symmetric matrices are studied.In the third chapter, we study the solving problem of the matrix equation AX = Bfor D symmetric matrices. We present the su?cient and necessary conditions of thesolvability for the linear restriction problem and the expressions of the solutions when the equation is consistent. Moreover we present the general expressions of the relatedleast squares solution and the related optimal approximation solution when the equationis inconsistent. At the same time we give a numerical algorithm and some numericalexamples to find the optimal approximation solution.In the fourth chapter, we study the solving problem of the matrix equation AX = Bfor D antisymmetric matrices. We present the suffcient and necessary conditions ofthe solvability for the linear restriction problem and the expressions of the solutionswhen the equation is consistent. Furthermore we present the general expressions of theleast squares solutions and the related optimal approximation solution when the equationis inconsistent. At the same time we give a numerical algorithm and some numericalexamples to find the optimal approximation solution.This thesis is supported by the National Natural Science Foundation of China(10571047).