This paper explores the design of hybrid digital and analog beamforming for large-scale antenna arrays, particularly for millimeter wave (mmWave) communication systems. The authors address the challenge of high cost and power consumption associated with conventional fully digital beamforming methods, which require one RF chain per antenna element. They propose a hybrid beamforming architecture that combines a low-dimensional digital beamformer with an RF beamformer implemented using analog phase shifters. The key contribution is that this hybrid structure can achieve the same performance as a fully digital beamformer with significantly fewer RF chains. Specifically, the paper shows that if the number of RF chains is twice the number of data streams, the hybrid beamforming structure can realize any fully digital beamformer, regardless of the number of antenna elements. For cases with fewer RF chains, the paper considers hybrid beamforming design for both point-to-point MIMO systems and downlink MU-MISO systems, proposing heuristic algorithms to maximize spectral efficiency under power constraints. The algorithms are further modified to handle finite resolution phase shifters, demonstrating effectiveness even with low-resolution phase shifters. Numerical simulations validate the proposed schemes, showing close performance to fully digital solutions.This paper explores the design of hybrid digital and analog beamforming for large-scale antenna arrays, particularly for millimeter wave (mmWave) communication systems. The authors address the challenge of high cost and power consumption associated with conventional fully digital beamforming methods, which require one RF chain per antenna element. They propose a hybrid beamforming architecture that combines a low-dimensional digital beamformer with an RF beamformer implemented using analog phase shifters. The key contribution is that this hybrid structure can achieve the same performance as a fully digital beamformer with significantly fewer RF chains. Specifically, the paper shows that if the number of RF chains is twice the number of data streams, the hybrid beamforming structure can realize any fully digital beamformer, regardless of the number of antenna elements. For cases with fewer RF chains, the paper considers hybrid beamforming design for both point-to-point MIMO systems and downlink MU-MISO systems, proposing heuristic algorithms to maximize spectral efficiency under power constraints. The algorithms are further modified to handle finite resolution phase shifters, demonstrating effectiveness even with low-resolution phase shifters. Numerical simulations validate the proposed schemes, showing close performance to fully digital solutions.