In this paper, we propose a transmit covariance optimization method to maximize the energy efficiency (EE) for the single-user distributed antenna system, where both the remote access units (RAUs) and the user are equipped with multiple antennas. Unlike previous related works, both the rate requirement and RAU selection are taken into consideration. Given this setup, we first propose an optimal transmit covariance optimization method to solve the EE optimization problem under fixed set of active RAUs and then the active RAU selection algorithm. For the transmit covariance optimization method, we split this problem into three subproblems, i.e., rate maximization problem, EE maximization problem without rate constraints, and power minimization problem. Then, a novel distance-based RAU selection method is proposed to determine the optimal set of active RAUs. Simulation results show that the performance of the proposed RAU selection is almost identical to the optimal exhaustive search method with significantly reduced computational complexity, and significantly outperforms the existing EE optimization methods.