The Perturbed Discrete Lyapunov Matrix Equation

This thesis is devoted to the Lyapunov matrix equation of discrete, which consists of five different chapters.In chapter 1, the background and present conditions are introduced and summaried for the study to the perturbation theory of matrix equations.In chapter 2, we give a new numerical iterative method for Lyapunov matrix equations, which is called Kronecker products iterative method. By means of the Kronecker products we get an iterative convergent sequence from the Lyapunov matrix equations.In chapter 3, we deal with the problems of eigenvalue estimation for the solution to the perturbed matrix Lyapunov equation. We obtain some eigenvalue inequalities on condition that X is a positive semidenite solution to the equation AT XA-X=-Q, which can be used in control theory and linear system stability.In chapter 4, we discuss the condition numbers for perturbed discrete Lyapunov matrix equation. We obtain the condition number inequality of the equation AT XA-X=-Q, which A and Q in this equation are stable.In chapter 5, the backward error of the solution to the perturbed discrete matrix Lyapunov equation is studied. The backward error bounds of the approximation positive definite solution to the equation are presented by using the property of matrix Kronecker products and matrix norms. The stability of above results is shown by a numerical example.