This chapter addresses the matrix-weighted consensus problem for homogeneous linear time-invariant dynamics. Although the agents share similar dynamics, their trajectories depend on their initial conditions and may differ significantly. By designing an appropriate consensus input, the agents in the network are expected to synchronize their outputs or even their state trajectories. Such synchronization algorithms not only provide theoretical insights into collective behaviors displayed in insects and animals but are also applicable to engineering systems, such as earthquake-resistant buildings, floating water plants, and cooperative synthetic aperture radars. In scalar-weighted graphs, the synchronization problem can be decoupled into n stabilization problems, each corresponding to an eigenvalue of the Laplacian matrix. However, in matrix-weighted networks, this decoupling strategy is not feasible because the agents’ system matrices do not commute with the matrix weights. While direct matrix-weighted synchronization algorithms can be designed, they often lead to large linear matrix inequalities that involve all system matrices. To address these challenges, this chapter proposes observer-based matrix-weighted consensus algorithms for both leaderless and leader–follower configurations in homogeneous linear time-invariant systems. Additionally, an adaptive gain tuning law is introduced to eliminate the need for global information, such as the smallest positive eigenvalue of the matrix-weighted Laplacian.