On The Structure Of Conjugate Symplectic Matrices And Constraint Equation Over Quaternion Field

Symplectic matrices are widely applied in mechanics,optics,modern geometry,control theory and cryptography design,it is an important tool to solve Hamilton eigenvalue problem effectively.At present,the research on the symplectic matrix is mainly about its properties,applications and various generalizations,but the symplectic solution of the linear system has not been reported in literature,and the related results are limited to the complex number domain.In order to expand the scope of research,the definition of conjugate symplectic matrix on four variables is given in this paper.The characteristic structure of the conjugate symplectic matrix class is studied.The necessary and sufficient conditions for the solution of the conjugate symplectic matrix or the three diagonal matrix of the three classes of linear systems are given and the solution is expressed.The specific contents are as follows:1.The symplectic matrix concept is generalized to the four element number body to form the conjugate symplectic matrix class,and then the eigenstructure and representation form of the positive definite symplectic matrix,the self conjugate symplectic matrix and the three diagonal symplectic matrix are characterized by the matrix four partitions,and a characteristic configuration algorithm for the conjugate symplectic matrix is given.2.The sufficient and necessary conditions for the two class of linear systems AS = B and ASB =C of the conjugate symplectic matrix and self conjugate symplectic matrix are discussed,and the correctness and feasibility of the method are tested by numerical examples.3.The sufficient and necessary conditions for the solution of the Sylvester equation AX+XB=C with three diagonal and self conjugate diagonal matrix solutions and the expression of the solution are discussed,The numerical example is used to test the correctness and feasibility of themethod.