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This paper presents an analytical performance investigation of transmit beamforming (BF) systems in Rayleigh product multiple-input multiple-output (MIMO) channels. We first derive new closed-form expressions for the cumulative distribution function, probability density function, and moments of the maximum eigenvalue of a product of independent complex Gaussian matrices, which are used to provide a complete statistical characterization of the received signal-to-noise ratio (SNR). We then derive a number of key performance metrics, including outage probability, symbol error rate, and ergodic capacity. We examine, in detail, three important special cases of the Rayleigh product MIMO channel: the degenerate keyhole scenario and the multiple-input single-output and single-input multiple-output scenarios, for which we derive insightful closed-form expressions for various exact and asymptotic measures (e.g., diversity order, array gain, and high SNR power offset, among others). We also compare the performance of transmit BF with orthogonal space-time block codes and quantify the benefit of exploiting transmitter channel knowledge in Rayleigh product MIMO channels. This is shown to be significant even for low-dimensional systems.