Two Systems Of Quaternion Matrix Equations

Matrix Equation Theory is one of the most important parts of Algebra Theory,and it plays a significant role on many fields such as Control Theory and Computational Algebra.As a strong mathematical tool to study the solvability of matrix equations,generalized inverse provides us a great convenience for our research.In this dissertation,we study on the solvability of two systems of linear matrix equations over quaternion division algebra by matrix equation theory and some knowledge of generalized inverse mainly,and give the general solution of these two systems respectively when solvability conditions are fulfilled.The dissertation is divided into five Chapters:In Chapter 1,we introduce the research background and progresses of quaternion division algebra,Moore-Penrose generalized inverse and matrix equation theory as well as the main work of this dissertation.In Chapter 2,we present some elementary knowledge of quaternion algebra and generalized inverse as well as several important lemmas used in proving the theorems of this dissertation.In Chapter 3,we study on the solvability of the system of quaternion matrix equa-tions as following A11XB11 =C11,A22XB22=C22,A33YB33=C33.and present the general solution to the system when all solvability conditions are satisfied.In Chapter 4,we study on the solvability of the more complicated system of quater-nion matrix equations as following A1X = C1,YB1 = D1,A2Z = C2,ZB2 =D2,A3W= C3,WB3 = D3,A4WB4 = C4,A5X+ YB5 + A6ZB6+A7WB7=C5.and give the general solution to the system by compatibility method when it is solvable.In Chapter 5,we summarize the methods used in proving theorems and suspect that these methods can give help in proving the solvability to other systems of matrix equations and constituting their general solutions.