This paper presents a novel channel estimation and beamforming design for beyond diagonal reconfigurable intelligent surfaces (BD-RIS) in multi-antenna systems. The authors propose a least squares (LS) based channel estimation strategy that jointly designs the pilot sequence and BD-RIS matrix to achieve the minimum mean square error (MSE). They analyze the training overhead and derive the MSE of the LS estimator, showing that the proposed method theoretically achieves the minimum MSE. The paper also investigates two BD-RIS scenarios and proposes beamforming design algorithms based on the estimated channel. Simulation results demonstrate the effectiveness of the proposed channel estimation scheme and beamforming design algorithms, showing that more inter-element connections in BD-RIS improve performance but increase training overhead. The authors also generalize the proposed channel estimation scheme to different BD-RIS aided scenarios, including multi-user systems and BD-RIS with hybrid transmitting and reflecting modes. The paper concludes that the proposed channel estimation method and beamforming design algorithms are effective for BD-RIS aided communication systems, with a trade-off between channel estimation performance and data transmission performance. The results show that the proposed methods achieve satisfactory performance close to the perfect CSI cases, with a practical trade-off between channel estimation performance and rate performance. The paper also discusses the circuit complexity of BD-RIS and the trade-off between training overhead and data transmission performance. The authors propose a tile-based channel construction to reduce the complexity and training overhead of channel estimation, and they show that this approach can be generalized to different BD-RIS aided scenarios. The paper also discusses the beamforming design for BD-RIS with reflective and hybrid/multi-sector modes, showing that the proposed methods can be applied to these scenarios. The results show that the proposed methods achieve satisfactory performance in both scenarios, with a trade-off between channel estimation performance and data transmission performance. The paper concludes that the proposed channel estimation and beamforming design methods are effective for BD-RIS aided communication systems, with a trade-off between channel estimation performance and data transmission performance.This paper presents a novel channel estimation and beamforming design for beyond diagonal reconfigurable intelligent surfaces (BD-RIS) in multi-antenna systems. The authors propose a least squares (LS) based channel estimation strategy that jointly designs the pilot sequence and BD-RIS matrix to achieve the minimum mean square error (MSE). They analyze the training overhead and derive the MSE of the LS estimator, showing that the proposed method theoretically achieves the minimum MSE. The paper also investigates two BD-RIS scenarios and proposes beamforming design algorithms based on the estimated channel. Simulation results demonstrate the effectiveness of the proposed channel estimation scheme and beamforming design algorithms, showing that more inter-element connections in BD-RIS improve performance but increase training overhead. The authors also generalize the proposed channel estimation scheme to different BD-RIS aided scenarios, including multi-user systems and BD-RIS with hybrid transmitting and reflecting modes. The paper concludes that the proposed channel estimation method and beamforming design algorithms are effective for BD-RIS aided communication systems, with a trade-off between channel estimation performance and data transmission performance. The results show that the proposed methods achieve satisfactory performance close to the perfect CSI cases, with a practical trade-off between channel estimation performance and rate performance. The paper also discusses the circuit complexity of BD-RIS and the trade-off between training overhead and data transmission performance. The authors propose a tile-based channel construction to reduce the complexity and training overhead of channel estimation, and they show that this approach can be generalized to different BD-RIS aided scenarios. The paper also discusses the beamforming design for BD-RIS with reflective and hybrid/multi-sector modes, showing that the proposed methods can be applied to these scenarios. The results show that the proposed methods achieve satisfactory performance in both scenarios, with a trade-off between channel estimation performance and data transmission performance. The paper concludes that the proposed channel estimation and beamforming design methods are effective for BD-RIS aided communication systems, with a trade-off between channel estimation performance and data transmission performance.