Inverse Eigenvalue Problem For Diagonal Matrix With Function Relation

In the higher algebra,the eigenvalue problem of the matrix is mainly to find the eigenvalue or eigenvector under the premise of a given matrix.And the inverse problem of the matrix eigenvalue we studied is to derive the prototype of the matrix under the premise of the eigenvalue or eigenvector of a given matrix.The research and exploration of the inverse problem of matrix eigenvalue is an important subject of numerical algebra,which is not only of theoretical significance,but also widely used in many disciplines(quantum mechanics,vibration mechanics,acoustics,optics,etc.).In vibration mechanics,for instance,the frequency of the discrete system is regarded as the eigenvalue of the matrix,and the modal is regarded as the eigenvector.The problem of exploring original vibration system can be transformed into the inverse problem of the matrix eigenvalue.In this paper,the diagonal matrix with the function relation is that there is a function relation among the diagonal elements of the diagonal matrix.Here,we discuss the inverse problem of eigenvalue of the trigonometric matrices with the function relation,the generalized Jacobi matrices and its sub-matrices.We further generalize the inverse problem of the eigenvalue of the four diagonal matrices with function relations,as follows:The first chapter introduces the development process and research status of inverse problem of matrix eigenvalue,and introduces the research object and content of this paper.In the second chapter,we mainly express the sufficient conditions and the expressions of the solution for the inverse problem of the eigenvalue of the three-diagonal matrix with function relations and the claw-type matrix.The conclusive theorem is given,and the algorithm of the solution is given.And the numerical examples are used to verify the accuracy and validity of the algorithm.In the third chapter,we discuss the sufficient conditions and the expressions of its solution for the inverse problem of the eigenvalue of the generalized Jacobi matrices with functional relations and its sub-matrices.We draw the conclusive theorem by deriving and also give the solutions to the problem of the algorithm,and finally give numerical examples to test.The fourth chapter is the generalization of the first two chapters.On the basis of the three diagonal matrices with functional relations,it is boldly speculated whether the four diagonal matrices can also get the expression of its solution by deriving.We construct the four diagonal matrix under given the function relations of its diagonal elements,so then we get the conclusive theorem by calculating and give the numerical examples to test.