The Structural Solutions Of Two Kinds Of Quaternion Matrix Equations And Their Optimal Approximation

With the development of science and technology,quaternion matrix has been widely used in many fields,such as spacecraft attitude control system,signal compression perception,cryptology and so on.Thereout,various constraint matrix equation problems are generated,which have become one of the most active and popular research topics in the field of computational mathematics.Constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set satisfying certain conditions.Different constrained conditions or different matrix equations lead to new constrained matrix equation problems.In this dissertation,the two kinds of quaternion matrix equations discussed are the Sylvester equation AX-XB=C and the more generalized equation AXB+CXD=E.The main purpose of the dissertation is to study the existence of constrained solutions and the optimal approximation problem of the two kinds of quaternion matrix equations in the space of several kinds of structural matrices,such as arrowhead matrix,Toeplitz matrix,conjugate symplectic matrix,M self-conjugate matrix and conjugated extended matrix.The main contents of the dissertation are as follows:In the first chapter,we will give the research background,the present situation and the development trend of the problems of constrained matrix equations,illuminate the main content for intensive discussions of this thesis.As propaedeutics,the operation properties of related complex matrix and the quaternion matrix as well as lemmas will be provided.In the second chapter,we discuss the arrowhead matrix and Toeplitz matrix solutions of Sylvester equation AX-XB=C and the optimal approximation problem over quaternion field.By using the real representation of a quaternion matrix,the Kronecker product of matrices and the specific structure of arrowhead matrix and Toeplitz matrix,the quaternion equation with constraints can be converted to an unconstrained equation over real number field.Then the necessary and sufficient condition for the existence of this matrix equation with arrowhead matrix and Toeplitz matrix and their general solution expression are obtained.At the same time,in the corresponding solution set,that the equation and the given quaternion matrix have the best approximate solution with minimal frobenius norm is derived.In the third chapter,the sufficient and necessary conditions and the expression of the solution of the quaternionic linear systems AXB+CXD=E with conjugate(or self-conjugate)symplectic matrix solutions and M self-conjugate matrix are discussed.In the meantime the numerical example is used to test the correctness and feasibility of the given method.In the fourth chapter,we discuss the column and row conjugated extended matrix solutions of quaternion equation AXB+CXD=E.Making use of the complex and real representations of a quaternion matrix,and the structural features of a conjugated extended matrix,the quaternion equation with constraints can be converted to an unconstrained equation.Then the necessary and sufficient condition for the existence of the quaternion matrix equation AXB+CXD=E with column and row conjugated extended matrix and their general solution expression are obtained.In the fifth chapter,we make a brief summary for this dissertation and introduce our future work.