Some Investigations On Matrix Equations And Operator Equations As Well As Generalized Inverses

In this dissertation, we establish the formulae of the extreme ranks of a quaternion matrix expression subject to some different generalized inverses. We also discuss some general solutions (with special structure) to some quaternion matrix equations. Moreover, we investigate the general solution with special structure to the adjointable operators equations over Hilbert C*-modules. These results further enrich and develop the quaternion matrix algebra and the operator algebra.The dissertation is divided into 6 chapters. In chapter 1, the research back-ground as well as the main contributions of this dissertation are introduced. Some preliminaries used in this paper are also presented. In chapter 2, we derive the ex-treme ranks of the quaternion matrix expression A-A3X1B3-A4X2B4 subject to some different generalized inverses. Using the results, we give some applications. In chapter 3, we give some solvable conditions for the the existence of solution to the system of quaternion matrix equations A1X=C1,A2X=C2, A3XB3= C3,A4XB4=C4. Moreover, by rank, we discuss the orthogonality of the so-lutions to the systems of quaternion matrix equations A1X1=C1,X1B1=D1 and A2X2=C2, X2B1=D2. In chapter 4, we consider the (skew)Hermitian solution to the system of quaternion matrix equations A1X1=C1,X1B1= D1,A2X2=C2,X2B2=D2, A3X1A3*+A4X2A4*=C5. As applications, we consider the (skew)bisymmetric, (skew)Hermitian reflexive solutions to the sys-tem of quaternion matrix equations AaX=Ca, XBa=Da, AbXAb*=Cb. In chapter 5, by investigating the general solution to the system of quater-nion matrix equations A1X=C1,XB1=D1, A3XB3±(A3XB3)*=C3, we consider the (skew)Hermitian anti-reflexive solution to the system of quater-nion matrix equations AaX=Ca,XBa=Da,AbXAb*=Cb. In chapter 6, we investigate the general common (skew)Hermitian solution to the equations A1X=C1,XB1=C2,A3XA3*=C3,A4XA4*=C4 for adjointable operators between Hilbert C*-modules.