Research On A Few Class Structure Of Quaternion Matrix Inverse Eigenvalue Problem

The problems of solving matrix inverse eigenvalue have broad application background in structure design, parameter identification, automatic control, quantum mechanics, spectroscopy and some other fields, so the study of inverse eigenvalue problem has become a hot topic for numerical algebra. With wide application of the quaternion, the certain structure of the quaternion matrix inverse eigenvalue problem has high theoretical and application value. In this paper, we study the solution of matrix inverse eigenvalue of conjugate symplectic matrix self-conjugate circulant matrix and tridiagonal matrix over the quaternion field. The thesis is divided into five chapters:In chapter 1, we give the brief introduction of the research background, present situation and development trends of the matrix inverse eigenvalue problem over the complex field and point out the further discussion in this paper. As preliminary knowledge, we introduce properties and related lemmas of the complex and quaternion field and so on.In chapter 2, we discuss the inverse eigenvalue problem of conjugate symplectic matrix over quaternion field. Firstly, we give the definition of conjugate symplectic matrices over quaternion field and discusse properties and characteristics of this matrix. We give the expression of solution of the inverse eigenvalue problem about certain conjugate symplectic matrix. Then, discusse sufficient and necessary conditions for the one class of conjugate symplectic matrix solution of the equation AS=B and give its expression.In chapter 3, discusse the inverse eigenvalue problem of the self-conjugate circulant matrix over quaternion field. First, for a given self-conjugate circulant quaternion matrix, according to the special structure of self-conjugate circulant quaternion matrix, establish the complex representation of A, so that we can get the eigenvalue expression of A. Then, for the given real numbers(?_i}_i=0^n-1, according to the contrary principle, we can get the solution of the inverse eigenvalue problem of the self-conjugate circulant matrix, and give specific algorithm of the solution.In chapter 4, we discuss tridiagonal matrix inverse eigenvalue problem on quaternion field. First of all, with the special structure of normal tridiagonal matrix over quaternion field, we can give the solution of this inverse eigenvalue problem. Then, via inverse Q-Arnoldi algorithm, we obtain the expression of the solution of inverse eigenvalue of positive definite tridiagonal matrix.In chapter 5, we make a brief summary for this paper and introduce our future work.