Investigation Of The Solutions To A System Of Quaternion Matrix Equations AX=B, XC=D

In this dissertation,we investigate the(P,Q)-(skew)symmetric solution,reflexive re-positive(semi)definite solution and reflexive re-positive per-(semi)definite solution to the system of quaternion matrix equations AX=B,CX=D.The dissertation is divided into 4 chapters.In Chapter 1,we introduce the research background and progresses as well as the work we have derived in this dissertation.Some preliminary knowledge used in this paper are also presented,e.g.,the basic matrix theory,symmetric matrix,the generalizcd inverses of matrices,some rank equalities of some matrix expressions, etc.In Chapter 2,we mainly investigate the following three problems:Problem 1 Give necessary and sufficient conditions for the existence of and the expressions for the(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system of quaternion matrix equations AX=B,XC=D.Problem 2 Find out the formulas of maximal and minimal ranks of(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system.Derive the expressions of the(P,Q)-symmetric and(P,Q)-skewsymmetric solutions with maximal and minimal ranks to the system mentioned above.Problem 3 IfΩ=φwhereΩis the set of all solutions of Problem 1,given E∈H^m×n,find X₀∈Ωsuch that‖X₀-E‖=(?)‖X-E‖where P and Q are all Hermitian involutions.By using the matrix techniques and matrix theory constructed in chapter 1, we first give a practical method to represent an involutory quaternion matrix and establish a representation for a(P,Q)-symmetric(or(P,Q)-skewsymmetric) matrix. Then,we discuss Problem 1,i.e.,establish necessary and sufficient conditions for the existence of and expressions for(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system.Next,we give formulas of extremal ranks of(P,Q)-symmetric and(P,Q)-skewsymmetric solutions to the system and present the(P,Q)-(skew)symmetric solution with extremal ranks to the system.After that,we investigate Problem 3,and give the expression of its solution.We also present a numerical example to illustrate the results.In Chapter 3,we first give a criterion for a partitioned quaternion matrix to be re-positive(semi)definite,then present a criterion for a quaternion matrix to be reflexive re-positive(semi)definite.Next,we establish a necessary and sufficient condition for the existence of re-positive(semi)definite solution to the system of quaternion matrix equations AX=B,XC=D as well as an expression of the general solution.Based on these results,we establish a necessary and sufficient condition for the existence of and an expression for reflexive re-positive(semi)definite solution to the system mentioned above.In Chapter 4,we consider the reflexive re-positive per-(semi)definite solution to the system of quaternion matrix equations AX=B,CX=D.In this Chapter, we derive a necessary and sufficient condition for the existence of and an expression for the reflexive re-positive per-(semi)definite solution to the system of quaternion matrix equations AX=B,XC=D.In addition,we give a necessary and sufficient condition for the existencc of and an expression for the re-postive per-(semi)definite solution to the system.The criteria for a partitioned quaternion matrix to be re-positive per-(semi)definite and a quaternion matrix to be reflexive re-positive per-(semi) definite are also established.