This paper addresses the design of linear precoders for fixed multiple input multiple output (MIMO) receivers, focusing on two design criteria: minimizing transmitted power subject to signal-to-interference-plus-noise ratio (SINR) constraints, and maximizing the worst-case SINR subject to a power constraint. The authors show that both problems can be solved using standard conic optimization packages. They derive optimality conditions for both problems and propose simple fixed-point iterations to find the solutions. The paper also explores the relationship between the downlink and uplink duality in the context of joint downlink beamforming and power control. The proposed precoder design is general and can be applied to symmetric systems, where it achieves the performance of minimum mean squared error (MMSE) receivers using simple matched filters. The paper includes a detailed review of conic optimization, including Second Order Cone Programs (SOCP), Semi-Definite Programs (SDP), and Generalized Eigenvalue Programs (GEVP). The feasibility of the power optimization problem is analyzed, and a conic optimization solution is provided. The KKT optimality conditions are derived, and two alternative methods for finding the optimal solution are presented: a fixed-point iteration and a dual SDP program. The paper concludes with a discussion on the robustness of the proposed algorithms to the rank of the effective channel and their performance in symmetric systems.This paper addresses the design of linear precoders for fixed multiple input multiple output (MIMO) receivers, focusing on two design criteria: minimizing transmitted power subject to signal-to-interference-plus-noise ratio (SINR) constraints, and maximizing the worst-case SINR subject to a power constraint. The authors show that both problems can be solved using standard conic optimization packages. They derive optimality conditions for both problems and propose simple fixed-point iterations to find the solutions. The paper also explores the relationship between the downlink and uplink duality in the context of joint downlink beamforming and power control. The proposed precoder design is general and can be applied to symmetric systems, where it achieves the performance of minimum mean squared error (MMSE) receivers using simple matched filters. The paper includes a detailed review of conic optimization, including Second Order Cone Programs (SOCP), Semi-Definite Programs (SDP), and Generalized Eigenvalue Programs (GEVP). The feasibility of the power optimization problem is analyzed, and a conic optimization solution is provided. The KKT optimality conditions are derived, and two alternative methods for finding the optimal solution are presented: a fixed-point iteration and a dual SDP program. The paper concludes with a discussion on the robustness of the proposed algorithms to the rank of the effective channel and their performance in symmetric systems.