Inverse Eigenvalue Problem For Some Snow-Like Matrices

Inverse eigenvalue problem for matrices is to solve matrices by some or all given eigenvalues and eigenvectors, and many inverse eigenvalue problems often provide some new restrictive conditions according to the actual requirement. For the research of inverse eigenvalue problem, it is of necessity and significance for theory of numerical algebra, and also imposes great influence on the resolution of practical application problem. Nowadays, it has been applied to different fields, such as quantum mechanics, solid mechanics, acoustics, optics, parameter identification, architectural engineering, vibration control and so on. The special snow-like matrix of the thesis means a kind of special matrix, whose two diagonal elements aren’t all zero severally, and elements lying the column and the row of node of two diagonal elements also aren’t all zero, but other elements all are zero. That is, nonzero elements in the matrix form the shape of Chinese characters "mi", so it is called snow-like matrix in English. The thesis is concerned with inverse eigenvalue problem and generalized inverse problem for some snow-like matrices. The some snow-like matrices can be divided into special snow-like matrices and general snow-like matrices. By using some methods of linear algebraic equations, recursion and assumption, we present the solvability and the uniqueness of questions, and give the expressions of the solutions. Finally numerical examples show feasibility of algorithm. The thesis consists of three chapters:The first chapter is prologue part. The prologue mainly divides into two parts, the first part dwells on practical significance and research situation about inverse eigenvalue problem for matrices, and introduces inverse eigenvalue problem for some snow-like matrices and its. present, the second part describes primary object of study and content structure.In chapter two, we study inverse eigenvalue problem and generalized inverse problem for two special snow-like matrices. The two special snow-like matrices are divided odd order snow-like matrices and even order snow-like matrices. We respectively discuss solvability of the two special matrices, the expression of the solution of the problem is given by deduction and calculation, and theorem is summarized. Lastly, numerical example is provided in order to prove validity of the algorithm.The third chapter researches inverse eigenvalue problem and generalized inverse problem for some generalized snow-like matrices. Those generalized snow-like matrices are obtained on the basis of special snow-like matrices of the chapter two. By using assumption, recursion and calculation, we give the conditions of existence and uniqueness of solutions, and summarize theorem for solving these problems. Lastly, we formulate concrete numerical example in order to validate validity of algorithm.