Some Investigations On The Solutions To Some Systems Of Quaternion Matrix Equations

In this dissertation, by using the fundamental theory and methods of gen-eralized inverse and rank of matrix, we investigate the least square solutions of some quaternion matrix equations subject to some equations. The least-norm and extremal ranks of the least square solutions mentioned above are discussed. Moreover, we characterize the submatrices in a solution to a system of quater-nion matrix equations. Some properties of the submatrices mentioned above are considered such as the extremal ranks, the uniqueness, the independence. As ap-plications, we also give-some necessary and sufficient conditions for the existence of some special partitioned solutions to the system.The dissertation is divided into 3 chapters.In Chapter 1, we introduce the research background and progresses of quater-nion, quaternion matrices and quaternion matrix equations as well as the work have been done in this thesis. Some preliminary knowledge of extremal rank formulas are also presented.In Chapter 2, we give the expression of the least square solutions of the linear quaternion matrix equation AXB= C subject to a consistent system of quaternion matrix equations D1X=F1,XE2=F2, and derive the maximal and minimal ranks and the least-norm of the above mentioned solutions.In Chapter 3, we suppose that the system of quaternion matrix equations A1X=C1,XB2=C2,A3XB3=C3, A4XB4=C4 has a solution X and partition X into a 2 x 2 block form. By using methods of rank of matrix, we investigate the extremal ranks,the uniqueness and the independence of submatrices in X. As applications, we also give some necessary and sufficient conditions for the existence of some special partitioned solutions to the system.