The self-excited vibration, such as galloping, dry friction induced vibration and so on, often occurs in many mechanical systems and structures. For several years, the effectiveness of a dynamic absorber for self-excited vibration has been known. However, the operating mechanism of a dynamic absorber for self-excited vibration is still poorly understood and a rational optimization procedure based on the operating mechanism is not established. In this paper, a simple and fundamental model with two degrees of freedom (DOFs), that is, a single-DOF system with negative damping in which one dynamic absorber is attached, is considered and the operating mechanism is investigated from the viewpoint of the energy balance. In order to clarify the operating mechanism, we apply a new type of complex modal analysis proposed by the authors to the system with negative damping, and the modal equations with the diagonalized mass, stiffness and damping matrices in the form of real second-order differential equations can be obtained. Using approximate solutions obtained from the modal equation, the energy generated due to negative damping and that dissipated due to positive damping can be estimated accurately. The results show that the appropriate decentralization of the excitation energy to each mode and the increase of the dissipation energy, caused by the dynamic absorber, play dominant roles in the stabilization of the system. In addition, the optimization procedure is formulated. The validity of the analytical results is verified by comparing with the numerical solution.