The Least-squares Problem Of General-symmetrizable Matrices And General-antisymmetrizable Matrices

The constrained matrix equation,least squares problems and the related optimal ap-proximation problems have been of interest in variety application in many fields. In-cluding particle physics and geology,the inverse problems of control theory,the inverseproblems of vibration theory, the approximation problem over finite elements and multi-dimensional space.We studied the following linear restriction problems,least squares problems and therelated optimal approximation problems for the general-symmetrizable matrices and thegeneral-antisymmetrizable matrices in this thesis.Problem I. Given X, B∈Rn×m,find A∈S such thatAX = B.Problem II .Given X, B∈Rn×m,find A∈S such thatAX ? B = min.Problem III. Given A?∈Rn×n,find A?∈S A such thatwhere S is a set of general-symmetrizable matrices or a set of general-antisymmetrizablematrices, S A is the solution set of problem I or the solution set of problem II,·is theFrobenius norm.The paper has four parts.In the first chapter, the background,the significance and progress situation for thestudy of inverse eigenvalues problems,linear restriction problems, the least squares prob-lems and the related optimal approximation problems are presented. And the main workof this paper is also simply introduced.In the second chapter, the properties of the general-symmetrizable matrices and thegeneral-antisymmetrizable matrices are studied.In the third chapter, we studied the solving problem of the equation AX = B forgeneral-symmetrizable matrix. We presented the su?cient and necessary conditions ofthe solvability for the linear restriction problem and the expressions of the solutions when the equation is compatible. Then we presented the general expressions of the relatedleast-squares solution and the related optimal approximation solution when the equationis not compatible. At the same time we gave the numerical algorithm and the numericalexamples to find the optimal approximation solution.In the fourth chapter, we studied the solving problem of the equation AX = B forgeneral-antisymmetrizable matrix. We presented the su?cient and necessary conditionsof the solvability for the linear restriction problem and the expressions of the solutionswhen the equation is compatible. Then we presented the general expressions of therelated least-squares solution and the related optimal approximation solution when theequation is not compatible. At the same time we gave the numerical algorithm and thenumerical examples to find the optimal approximation solution.This thesis is supported by the National Natural Science Foundation(10571047) ofChina.