  This letter presents optimal beamforming and outage analysis for a
Reconfigurable Intelligent Surface (RIS)-aided multiple input single output
downlink system under Rician fading on both the direct and the RIS-assisted
indirect links. We focus on maximizing the capacity for two transmitter
architectures: fully digital (FD) and fully analog (FA). This capacity
maximization problem with optimally configured RIS is shown to be $L_1$
norm-maximization with respect to the transmit beamformer. To obtain the
optimal FD beamformer, we propose a complex $L_1$-PCA-based algorithm whose
complexity is significantly lower than the existing semi-definite
relaxation-based solutions. We also propose a low-complexity optimal
beamforming algorithm to obtain the FA beamformer solution. Further, we derive
analytical upper bounds on the SNR achievable by the proposed algorithms and
utilize them to characterize the lower bounds on outage probabilities. The
derived bounds are numerically shown to closely match the achievable
performance for a low-rank channel matrix and are shown to be exact for a
unit-rank channel matrix.
