Future multi-input multi-output (MIMO) wireless communications systems will use beamforming as a first-step towards realizing the capacity requirements necessitated by the exponential increase in data demands. The focus of this work is on beam alignment for time-division duplexing (TDD) systems, for which we propose a number of novel algorithms. These algorithms seek to obtain good estimates of the optimal beamformer/combiner pair (which are the dominant singular vectors of the channel matrix). They are motivated by the power method, an iterative algorithm to determine eigenvalues and eigenvectors through repeated matrix multiplication. In contrast to the basic power method which considers only the most recent iteration and assumes noiseless links, the proposed techniques consider information from all the previous iterations of the algorithm and combine them in different ways. The first technique (Sequential Least-Squares method) sequentially constructs a least-squares estimate of the channel matrix, which is then used to calculate the beamformer/combiner pair estimate. The second technique (Summed Power method) aims to mitigate the effect of noise by using a linear combination of the previously tried beams to calculate the next beam, providing improved performance in the low-SNR regime (typical for mmWave systems) with minimal complexity/feedback overhead. A third technique (Least-Squares Initialized Summed Power method) combines the good performance of the first technique at the high-SNR regime with the low-complexity advantage of the second technique by priming the summed power method with initial estimates from the sequential method.