Inverse Eigenvalue Problem For Generalized Diagonal Matrices

Source of inverse eigenvalue problem is very extensive, which has been applied to solid mechanics, physics, quantum mechanics, automation, and many other fields. Thus, it has a strong physical background and high academic value.This article introduces the definition, natures, theorems and algorithms to the inverse problem, the eigenvalue problem and the inverse eigenvalue problems for real diagonal matrices. At the same time, search and absorb a large number of inverse problems associated with the research results. On this basis, discussed the eigenvalue problem generalized diagonal matrix. The article includes four aspects as follows:The first, It describes the eigenvalue problem and the inverse eigenvalue problems for three-diagonal matrices, introduces the application background and current developments of such issues. Focused on inverse eigenvalue problem for principal square submatrix of generalized Jacobi matrices. The existence and uniqueness of the question A's solution is presented in the form of a theorem, and A numerical example verifies effectiveness of the algorithm.The second, It describes the concepts and results of the inverse eigenvalue problems for real symmetric banded matrix, introduces several typical inverse eigenvalue problem for five-diagonal matrix. The most important thing is to use two different methods to study inverse eigenvalue problem for real five-diagonal matrices with proportional relation. Discussed the only existing conditions of the solution for question B1 and question B2, Expressions of the solution are given and numerical examples are given to checking separately.The third, It describes the concepts, related solutions of questions and algorithms on inverse eigenvalue problem for generalized periodic Jacobi matrix. In this chapter focuses on inverse eigenvalue problem for sub-periodic generalized Jacobi matrices. Discussed the conditions of a unique solution for question C, given the expression of the solution, and provided a numerical example.The forth, It Investigates future research directions of inverse eigenvalue problem for generalized diagonal matrices from four aspects of theory, algorithms, matrix type, and constraints separately.