The Study Of Iterative Solutions For Several Linear Matrix Equations And Its Application

The status of linear matrix equations is significant in linear systems theory.Actually, its theory can be widely used in areas of control theory, such as parame-ter identification, structure design, vibration theory, linear optimum control. Theresearch of solving the linear matrix equations not only can extend and developthe matrix theory but also can provide the theoretical intention and the practicalbasis. Thus the theory has high value and practical significance.Illuminated by conjugate gradient method, we construct new iterative al-gorithm to solve three class of linear matrix equations from different aspects,respectively. We show the numerical solution of the matrix equations, prove theconvergence of the proposed method by matrix norms and trace and make thecomparison with some already existing results. With this, we give some conclu-sions about the applications. Some numerical results show the effectiveness andsuperiority. Main contents as follows:In chapter one, we first present some background knowledge and recent worksfor linear matrix equation and its solution. Then we propose the status quo ofwhich is the main work and give some basic symbols and definitions used in ourthesis.In chapter two, we study generalized Sylvester matrix equation obtainedfrom the design of eigenstructure assignment. Illuminated by conjugate gradientmethod for solving linear equations, we solve the equation with the method of it-eration and give its application in the descriptor system. Some numerical resultsshow the effectiveness of the proposed approach and the stability of the system.In chapter three, the general linear matrixequation is studied. By deformation for the proposed iterative method, we con-struct the suitable iterative algorithm and propose the consistent solution. Fur-ther, the inconsistent problems are studied by equations transformation and theleast-square solutions are also proposed. The results extend some already existingrelative results to the more general situation by comparison for applications.In chapter four, , the consistent dual variant ma-trix equation is studied. By construct an iterative method, we obtain the solutionand the least norm solution and also study the optimal approximation solutionpair to a given matrix pair. Some numerical results show the effectiveness of the proposed approach.