The Inverse Eigenvalue Problems Of Symmetric Arrow-Head Matrix

The inverse eigenvalue problems and the inverse problems of matrix are widely applied in many fields,such as solid mechanics,particle physics,quantum mechanics,structure design,system parameter identification and automation etc.The symmetric arrow-head matrix is important in many aspects.In the thesis,we mainly discuss three problems:inverse eigenvalue problems of symmetric arrow-head matrices,generalized inverse eigenvalue problems for symmetric arrow-head matrix pair and inverse problems of symmetric arrow-head matrices,which are described as follows:ProblemⅠ:Given partial submatrices of a symmetric arrow-head matrix A,its partial eigenvalues and partial elements or all elements of corresponding eigenvectors, find the symmetric arrow-head matrix A and the unknown elements of the corresponding eigenvectors.ProblemⅡ:Given a positive-definite symmetric arrow-head matrix B and partial generalized eigenvalues,corresponding generalized eigenvectors of a matrix pair(A,B) ,find the symmetric arrow-head matrix A.ProblemⅢ:Given two vectors(or two matrices)X,Y,find the symmetric arrow-head matrix A such that AX=Y or‖AX-Y‖=min.The main researchful results of the thesis are as below:1.With respect to Problem I,we consider properties for eigenvalues and eigenvectors of symmetric arrow-head matrices.According to them,we study three kinds of inverse eigenvalue problems of symmetric arrow-head matrices,obtain the necessary and sufficient conditions for the problems to have unique solutions,and give the three effective algorithms for the problems respectively.2.Concerning ProblemⅡ,We consider the properties for generalized eigenvalues and eigenvectors of a symmetric arrow-head matrix pair(A,B).Basing of them, we study two generalized inverse eigenvalue problems,and deduce the necessary and sufficient conditions from the two problems to have unique solutions.In addition.The corresponding algorithms are given to compute the unique solutions for the problems respectively.3.About ProblemⅢ,we first study a kind of the inverse problem for the symmetric arrow-head matrices,and obtain the necessary and sufficient conditions under which they have a unique solution and give an efficient algorithm to compute the solutions.Secondly we consider two kinds of the least-squares problems for inverse problems of symmetric arrow-head matrices,discuss the conditions under which they have solutions,and derive the expressions of the least-squares solutions.The corresponding algorithms are presented to compute the least-squares solutions.