Hermite Positive Definite Solution of the Quaternion Matrix Equation X^m + B*XB = C  

作　　者：Yiwen Yao Guangmei Liu Yanting Zhang Jingpin Huang Yiwen Yao;Guangmei Liu;Yanting Zhang;Jingpin Huang(College of Mathematics and Physics, Guangxi Minzu University, Nanning, China)

机构地区：[1]College of Mathematics and Physics, Guangxi Minzu University, Nanning, China

出　　处：《Journal of Applied Mathematics and Physics》2023年第11期3760-3772,共13页应用数学与应用物理（英文）

摘　　要：This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X^m+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X^m+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .

关 键 词：QUATERNION Matrix Equation Hermite Positive Definite Solution Matrix Inequality ITERATIVE CONVERGENCE 

分 类 号：O17[理学—数学]