Research On The Structural Solutions Of Two Classes Of Quaternion Matrix Equations And Their Extremal Ranks

With the development of quaternion theory and research,the quaternion matrices were widely used in control systems,animation game design,system stability analysis,spacecraft positioning,quantum mechanics and other fields.Meanwhile,the problem raised extensive attention from scholars at home and abroad for solving quaternion matrix equations,and has become a hot topic in the field of matrix algebra.By using the algebraic structure of quaternion matrices,Moore-Penrose generalized inverse,and the rank of block-matrix,in this dissertation,the structural solutions of quaternion matrix equations [AX,XC] = [B,D] and[AX,XC,EXF] = [B,D,G],and their extremal ranks are discussed.The specific content is divided into 5 chapters,and summarized as follows:In Chapter 1,some basic information is presented,such as the research background,research status and development trend of quaternion matrix,main content,structural framework,basic concepts and some important lemmas.In Chapter 2,the extremal rank of complex matrices of general solution is discussed in the quaternion matrix equation [AX,XC] = [B,D].Then,when there is no solution,the extremal rank formula of the least square complex component solution is derived.In Chapter 3,the necessary and sufficient conditions for the existence of generalized Hamiltonian solutions of quaternion equations [AX,XC] = [B,D],and their expressions are discussed.Furthermore,the extremal rank formula of generalized Hamilton solution,the generalized Hamilton solution complex component and the least square generalized Hamilton complex component solution are given.In Chapter 4,the necessary and sufficient conditions for the existence of centrosymmetric matrix solution of quaternion matrix equations [AX,XC,EXF] = [B,D,G],and their expressions are discussed.Moreover,the extremal rank formula of the centrosymmetric solutions are obtained.In Chapter 5,a brief summary of this work is presented,and the future work is introduced.