This paper proposes a joint optimization framework for energy-efficient precoding and feeder-link-beam matching design in a multi-gateway multi-beam bent-pipe satellite communication system. The proposed scheme jointly optimizes the precoding vectors at the gateways and amplifying-and-matching mechanism at the satellite to maximize the system weighted energy efficiency under the transmit power budget constraint. The technical designs are formulated into a non-convex sparsity problem consisting of a fractional-form objective function and sparsity-related constraints. To address these challenges, two iterative efficient designs are proposed by utilizing the concepts of Dinkelbach's method and the compress-sensing approach. The simulation results demonstrate the effectiveness of the proposed scheme compared to another benchmark method.