Several Studies About The Symmetric Tridiagonal Matrix Inverse Generalized Eigenvalue Problems

Matrix inverse generalized eigenvalue problem(also called inverse generalized eigenvalue problem)concerns the reconstruction of matrices under certain constraint conditions which are from prescribed complete/partial information of generalized eigenvalues or eigenvectors and from partial submatrices or elements.The generalized inverse eigenvalue problem is widely involved in related fields such as mathematical physics,structural dynamics,molecular optics,etc.With the development of these fields,many different types of questions have been proposed to promote the rapid development of the theory of generalized inverse eigenvalue problems.However,as the inverse problem cherishes its own complexity,and the theory and the actual difference,it makes the study of matrix generalized inverse eigenvalue problem process is relatively slow,except for special cases such as symmetric tridiagonal matrix and periodic tridiagonal matrix inverse problem research,which mainly use multiple eigenvalues,or the cha-racteristics of the matrix or defect data,or principal submatrices,to construct the correspondinng matrix,but the corresponding theoretical results for inverse generalized symmetric tridiagonal eigenvalue problem by the non-principal submatrices and the defected generalized eigenpairs are not common.In this dissertation,two kinds of symmetric tridiagonal matrix inverse generalized eigenvalue problem such as Anα = λCnα have been considered,By splitting the matrix An into3×3block form,when Cn ∈ Rn×n is the diagonal matrix or symm-etric tridiagonal matrices,respectively,the corresponding inverse generalized problem has been studied.This dissertation includes four chapters,which is organized as follows:In Chapter 1,the corresponding background of research and the preliminary knowledge and the main work of this paper have been illustrated.In Chapter 2,the corresponding basic theory have been introduced.In Chapter 3,We first give the definition of the first kind of inverse generalized eigenvalue problem for symmetric tridiagonal matrix and then the solvable conditions of this problem and solution form have been presented.Finally,the corresponding examples are given.In the end,the definition of the second kind of inverse generalized eigenvalue problem for symmetric tridiagonal matrix has been proposed and then the solvable conditions of problem and solution form have been presented.Finally,the correspond-ing examples are given.