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Recently, a new type of device that enables reduced capability (RedCap) has been identified and standardized in the fifth generation (5G) New Radio (NR). Because RedCap user equipment (UE) is designed to have low device cost and complexity compared with high-end devices, single-layer transmission is mainly considered for RedCap UE. For single-layer transmission, traditional eigenmode beamforming, which transmits a data stream through the maximum eigenmode direction, is applicable for base stations (BSs), and a simple matched filter can be used on the UE side. Under this operation scenario, the performance of eigenmode beamforming is analyzed in this study, and the effect of the number of UE antennas on the achievable rate is investigated. The asymptotic achievable rate for eigenmode beamforming with practical channel training for both the uplink and downlink is derived as a closed-form expression for any number of UE antennas <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. For the analysis, the statistics of the maximum eigenvalue are derived using the Tracy-Widom distribution. The results of the analysis show that the maximum eigenvalue is a concave function in terms of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, and there is an optimal number of UE antennas <inline-formula> <tex-math notation="LaTeX">$n_{\mathrm {opt}}$ </tex-math></inline-formula> that maximizes the achievable rate. Simulations verify the analytical results and show that <inline-formula> <tex-math notation="LaTeX">$n_{\mathrm {opt}}$ </tex-math></inline-formula> maximizing the achievable rate decreases as the numbers of BS antennas and co-scheduled UEs increase.