This paper introduces a novel blind iterative projections algorithm to address challenges associated with codebook-based beamforming in 5G and beyond systems. In contrast to existing methodologies, our proposed algorithm eliminates quantization noise and search space scanning latency. The blind nature of our approach capitalizes on information-bearing symbols, yielding performance close to theoretical expectations. In contrast to the prevalent iterative least squares (LS) algorithms in the literature, our proposal does not necessitate the constant modulus or finite alphabet constraints on the desired signal. This makes our algorithm more general and allows us to demonstrate its stability analytically and capability to perform singular value decomposition (SVD) of the received signal matrix. Our algorithm achieves SVD optimality with lower computational costs than conventional SVD methods. We present an analysis of the Signal to Noise Ratio (SNR) at each iteration, elucidating the impact of batch size and noise variance on the SNR gain at convergence. We apply our proposed algorithm to receive-side hybrid beamforming and show that it offers superior performance compared to various existing approaches documented in the literature.