Researches On Simplifying Quaternion Matrices And Solving Quaternion Matrix Equations

Quaternion and quaternion matrix theory, which is widely used in geostatics, quan-tum mechanics, robotics, inertial navigation, gyroscope, satellite attitude control, com-puter graphics and so on, has been paid more and more attention by both of pure andapplied algebraist in recent years. The reducing of a quaternion matrix, the reducing of aquaternion matrix pair and the solutions of several quaternion matrix equations are dis-cussed in this dissertation. All of these are very important problems existing in the fieldof quaternion matrix theory and its applications. The content of the dissertation mainlyincludes the following results:1. In the second chapter, we study the general LU decomposition of quaternionmatrix. It is established that any quaternion matrix A exists general LU decomposition,and the canonical form of the general LU decomposition of A is unique. Further, basedupon this result, we obtain a necessary and su?cient condition for A to have a LUdecomposition. As an application, we also research the solution of the overdeterminedquaternion infinite linear system, give a su?cient condition for the existence of the leastsquares solution.2. In the third chapter, firstly, we give the definition of contragredient equivalenceof quaternion matrix pair. And then, the canonical form of quaternion matrix pair undercontragredient equivalence is obtained. Furthermore, we prove that, disregarding the orderof blocks, the canonical form of a given quaternion matrix pair is unique. Beside these,we also give a necessary and su?cient condition for two quaternion matrix pairs to becontragredient equivalent. Using these results, we study the solution of the system ofquaternion matrix equations3. An algorithm for the simultaneously complexification of quaternion matrix regularpair is given in the first part of the forth chapter. And then, based upon this algorithm, analgorithm for simultaneously complexification of any quaternion matrix pair is obtained.As an application, the quaternion matrix equationAXB - CXD = Eis studied, and an algorithm for computing the solutions of it is achieved. 4. In the forth chapter, we also investigate the solution of complex matrix equationAXB - CXD = E.Particularly, by considering the complex matrix equation as a quaternion matrix equation,we obtain necessary and su?cient conditions for existence of solution and existence ofunique solution respectively. Furthermore, an algorithm for solving the complex matrixequation is given.