Research On Two Kinds Of Constrained Matrix Equations Over The Quaternion Field

The problems of solving constrained matrix equations have been a hot topic in the field of numerical algebra in recent years, and have been importantly applied in many field such as structural design, parameter identifications, molecular spectra, nonlinear programming and dynamic analysis. In this paper, we study the solution of two kinds of constrained matrix equations over the quaternion field and their optimal approximation. The thesis is divided into five chapters:In chapter 1, we give the background, the present situation and the development trend of the problems of constrained matrix equations over the complex field, the main research of this thesis. And introduce the necessary preliminaries such as properties, related lemmas of the complex and quaternion field and so on.In chapter 2, discusses the D-Self-conjugate solution and least squares problem of the matrix equation AXB =C over quaternion field. Firstly, by using inner product of quaternion vectors along with a positive definite matrix D, the definition of D-self-conjugate matrix is given. Then, use Moore-Penrose inverse of a quaternion matrix, the necessary and sufficient conditions for the existence of D-Self-conjugate matrix solutions for AXB=C are obtained together with the general forms of such solutions. In addition, the least square D-Self-conjugate solutions for AXB=C has been derived by generalized singular value decomposition on the quaternion matrix pairs, and a numerical example demonstrates the feasibility of the proposed method.In chapter 3, we discuss the problem of the circulant matrix and the Hcirculant matrix solution of the unified algebraic Lyapunov equation* *A X +XA+qA XA = -P over quaternion field and its optimal approximation. First, by using the comple representation of a quaternion matrix and Kronecker product, some necessary and sufficient conditions for the circulant solution of the unified algebraic Lyapunov equation over quaternion field are derived. At the same time, the general forms of such circulant matrix solutions and its optimal approximation are obtained. Second, by using the iterative method, the unified algebraic Lyapunov equation over H- circulant matrix solution can be determined automatically. Finally, two numeral example shows the feasibility of the method.In chapter 4, we discuss the problem of the toeplitz matrix solution of the unified algebraic Lyapunov equation A*X +XA+qA*XA = -P over quaternion field and its optimal approximation. By using the comple representation of a quaternion matrix, the specific structure of a toeplitz matrix and transforming a constrained quaternion matrix equation into an unconstrained real matrix equation through Kronecker product, thus the necessary and sufficient condition for the existence of a toeplitz matrix solution and the general solution of the equation are obtained. Meanwhile, in the solution set, the optimal approximation solution which has minimal Frobenius norm for given quaternion toeplitz matrix is derived.In chapter 5, we make a brief summary for this paper and introduce our future work.