  Beyond-diagonal reconfigurable intelligent surface (BD-RIS) has been proposed
recently as a novel and generalized RIS architecture that offers enhanced wave
manipulation flexibility and large coverage expansion. However, the
beyond-diagonal mathematical model in BD-RIS inevitably introduces additional
optimization challenges in beamforming design. In this letter, we derive a
closed-form solution for the BD-RIS passive beamforming matrix that maximizes
the sum of the effective channel gains among users. We further propose a
computationally efficient two-stage beamforming framework to jointly design the
active beamforming at the base station and passive beamforming at the BD-RIS to
enhance the sum-rate for a BD-RIS aided multi-user multi-antenna
network.Numerical results show that our proposed algorithm achieves a higher
sum-rate while requiring less computation time compared to state-of-the-art
algorithms. The proposed algorithm paves the way for practical beamforming
design in BD-RIS aided wireless networks.
