In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the ta...In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .展开更多
In order to solve the linear algebraic system AX=b in complex domain, where A is a weakly cyclic of index p=3 matrix (p cyclic matrix), the convergence properties of SOR are studied in the paper. In section 1, we give...In order to solve the linear algebraic system AX=b in complex domain, where A is a weakly cyclic of index p=3 matrix (p cyclic matrix), the convergence properties of SOR are studied in the paper. In section 1, we give some definitions. In section 2, the necessary conditions for convergent complex SOR are given moreover the necessary and sufficient conditions in some special situations are also presented. In section 3, we expand the techniques applied by R.S. Varga et al., and it is established that the results of R.S. Varga et. al. are special cases of our work.展开更多
文摘In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .
文摘In order to solve the linear algebraic system AX=b in complex domain, where A is a weakly cyclic of index p=3 matrix (p cyclic matrix), the convergence properties of SOR are studied in the paper. In section 1, we give some definitions. In section 2, the necessary conditions for convergent complex SOR are given moreover the necessary and sufficient conditions in some special situations are also presented. In section 3, we expand the techniques applied by R.S. Varga et al., and it is established that the results of R.S. Varga et. al. are special cases of our work.
