Iterative Methods For The Reflexive Solution Of Several Quaternion Matrix Equations

Quaternion and quaternion matrix equations have important applications in rigid body kinematics, computer graphics, informatics, quantum mechanics, etc.Various solu-tions of quaternion matrix equations has become a popular research topic. The iterative method is a very important method to solve matrix equations and there have been many results. Due to the non-commutativity of quaternion multiplication, some well-known equalities for real and complex matrices no longer hold for quaternion matrices, which make the study of quaternion matrix equation much more complex than that of real and complex equation. And the iterative methods for real and complex matrix equations are no longer suitable for quaternion matrix equations. In this dissertation, we present it-erative methods for the reflexive solutions of several quaternion matrix equations. By using a real inner product space builded on quaternion matrices, we prove that the it-erative algorithms will automatically determine the solvability of the quaternion matrix equations over reflexive matrices, and when the matrix equations are consistent over reflexive matrices, reflexive solutions can be obtained within finite iteration steps. By using a real representation of quaternion matrices, we convert the problems of quater-nion matrix equations to the problems of real matrix equations, so as to avoid the the non-commutativity of quaternion multiplication. By studying the least Frobenius norm solutions to the corresponding real matrix equations, we prove that when some special initial matrices are selected, the algorithms can get the least Frobenius norm reflexive solutions of the quaternion matrix equations. Furthermore, the optimal approximate re-flexive solutions to given matrices can be derived.In chapter2, we present an iterative method for the reflexive solution of quater-nion matrix equation AXB+CXHD=F. By using a real inner product space builded on quaternion matrices, we prove the convergence of the algorithm. By using the real representation of quaternion matrices, we prove that the least Frobenius norm reflexive solution of the quaternion matrix equation AXB+CXHD=F can be derived by the iterative algorithm. And then, we obtain the optimal approximate reflexive solution of the quaternion matrix equation AXB+CXHD=F for a given reflexive quaternion ma-trix. The matrix equation plays important roles in the system theory. There have been some results on the matrix equation∑AlXBl+∑CsXTDs=F over real field. In chapter3, we present an iterative method for the reflexive solution of quaternion matrix equation and its optimal approxima-tion reflexive solution for a given reflexive quaternion matrix. Then we prove the con-vergence of the algorithm. The matrix equation includes some important matrix equations in matrix theory. There have been some results on the matrix equation over complex field. In chapter4, we present an iterative method for the generalized (P,Q)-reflexive solution of quaternion matrix equation and its optimal approximation generalized (P, Q)-reflexive solution for a given quaternion matrix. Then we prove the conver-gence of the algorithm. In chapter5, we propose an iterative method for the gener-alized (P,Q)-reflexive solution of quaternion matrix equations with j-conjugate of the unknowns s=1,2,... N and its optimal approximation generalized (P,Q)-reflexive solution for a given quaternion matrix group. Then we prove the convergence of the algorithm.The numerical examples in this dissertation show the efficiency of the presented algorithms. These results enrich the iterative methods for quaternion matrix equations.