This paper proposes a low-complexity hybrid analog/digital precoding algorithm for downlink multi-user millimeter wave (mmWave) systems. The algorithm combines analog and digital processing to achieve high sum rates while minimizing training and feedback overhead. The key contributions include: (1) a hybrid precoding/combining algorithm for multi-user mmWave systems, where the BS performs hybrid analog/digital precoding and the MSs use analog-only combining; (2) performance analysis in single-path and large-dimensional regimes; and (3) characterization of rate loss due to joint quantization of analog and digital codebooks. The algorithm is evaluated through simulations, showing that it achieves higher sum rates than analog-only beamforming and approaches the performance of unconstrained digital beamforming with small codebooks. The algorithm assumes limited feedback and is general for arbitrary array geometries. The analysis shows that the proposed algorithm outperforms analog-only solutions, especially in sparse mmWave channels with large antenna arrays. The results demonstrate that the hybrid approach effectively manages multi-user interference and achieves near-optimal performance with low complexity.This paper proposes a low-complexity hybrid analog/digital precoding algorithm for downlink multi-user millimeter wave (mmWave) systems. The algorithm combines analog and digital processing to achieve high sum rates while minimizing training and feedback overhead. The key contributions include: (1) a hybrid precoding/combining algorithm for multi-user mmWave systems, where the BS performs hybrid analog/digital precoding and the MSs use analog-only combining; (2) performance analysis in single-path and large-dimensional regimes; and (3) characterization of rate loss due to joint quantization of analog and digital codebooks. The algorithm is evaluated through simulations, showing that it achieves higher sum rates than analog-only beamforming and approaches the performance of unconstrained digital beamforming with small codebooks. The algorithm assumes limited feedback and is general for arbitrary array geometries. The analysis shows that the proposed algorithm outperforms analog-only solutions, especially in sparse mmWave channels with large antenna arrays. The results demonstrate that the hybrid approach effectively manages multi-user interference and achieves near-optimal performance with low complexity.