In this paper, we propose a transmit covariance optimization method to maximize the energy efficiency (EE) for a single-user distributed antenna system, where both the remote access units (RAUs) and the user are equipped with multiple antennas. Unlike previous related work, both the rate requirement and RAU selection are taken into consideration. Here, the total circuit power consumption is related to the number of active RAUs. Given this setup, we first propose an optimal transmit covariance optimization method to solve the EE optimization problem under a fixed set of active RAUs. More specifically, we split this problem into three subproblems, i.e., the rate maximization problem, the EE maximization problem without rate constraint, and the power minimization problem, and each subproblem can be efficiently solved. 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 the performance of the proposed algorithm significantly outperforms the existing EE optimization methods.