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We consider multi-input multi-output (MIMO) transmit beamforming under the uniform elemental power constraint. This is a nonconvex optimization problem, and it is usually difficult to find the optimal transmit beamformer. First, we show that for the multi-input single-output (MISO) case, the optimal solution has a closed-form expression. Then we propose a cyclic algorithm for the MIMO case which uses the closed-form MISO optimal solution iteratively. The cyclic algorithm has a low computational complexity and is locally convergent under mild conditions. Moreover, we consider finite-rate feedback methods needed for transmit beamforming. We propose a simple scalar quantization method, as well as a novel vector quantization method. For the latter method, the codebook is constructed under the uniform elemental power constraint and the method is referred as VQ-UEP. We analyze VQ-UEP performance for the MISO case. Specifically, we obtain an approximate expression for the average degradation of the receive signal-to-noise ratio (SNR) caused by VQ-UEP. Numerical examples are provided to demonstrate the effectiveness of our proposed transmit beamformer designs and the finite-rate feedback techniques.