Researches On Real And Complex Solutions To Quaternion Matrix Equations

In this dissertation,by studying the maximal and minimal ranks of the general solutions to certain systems of quaternion matrix equations,necessary and sufficient conditions for the existences and expressions of real and complex solutions to some systems of quaternion matrix equations are given.These results further enrich and develop the quaternion matrix algebra.The dissertation is divided into 5 chapters.In Chapter 1,we introduce the research background and progresses of quaternion,quaternion matrices,quaternion matrix equations,extremal ranks of matrix expressions,generalized inverse of matrix as well as the work we have done in this dissertation.Some preliminary knowledge used in this paper are also presented.In Chapter 2,we give the maximal and minimal ranks of the quaternion matrix expression C₄ -- A₄XB₄ subject to quaternion matrix equations A₁X = C₁,XB₂ = C₂,A₃XB₃ = C₃, ncccssary and sufficient conditions for the solvability of the quaternion matrix equations A₁X = C₁,XB₂ = C₂,A₃XB₃ = C₃,A₄XB₄ = C₄ are derived by rank equalities.In Chapter 3,we give the maximal and minimal ranks of the general solution to quaternion matrix equation AXB = C,and derive necessary and sufficient conditions for the existences and the expressions of real and complex solutions to the matrix equation.As an application,necessary and sufficient conditions for quaternion matrix equations A₁XB₁ = C₁,A₂XB₂ = C₂ to have real and complex solutions are presented.In Chapter 4,we investigate the maximal and minimal ranks of the general solution to quaternion matrix equations A₁XB₁ = C₁,A₂XB₂ = C₂,and derive necessary and sufficient conditions for the existences and the expressions of real and complex solutions to the matrix equations.As an application,necessary and sufficient conditions for quaternion matrix equations A₁XB₁ = C₁,A₂XB₂ = C₂,A₃XB₃ = C₃ to have real and complex solutions are given.In the last Chapter,we investi- gate the extremal ranks of the general solution to quaternion matrix equations A₁X = C₁,XB₂ = C₂,A₃XB₃ = C₃,and derive necessary and sufficient conditions for the existences and the expressions of real and complex solutions to the matrix equations.As an application,necessary and sufficient conditions for quaternion matrix equations A₁X = C₁,XB₂ = C₂,A₃XB₃ = C₃,A₄XB₄ = C₄ to have real and complex solutions are established.