This paper investigates the stabilizability and decentralized stabilization of multi-agent systems under matrix-weighted directed graphs. By introducing the concept of matrix-weighted independent strongly connected components and exploiting the decomposition of the block Laplacian matrix, a necessary and sufficient condition for system stabilizability is established. Under an additional assumption on the null-space structure of the Laplacian, a rank-based criterion that is easier to verify is further derived. For the case of decentralized state feedback, an analytical framework is developed to infer the stability of the overall closed-loop system from the stability of local closed-loop subblocks, and sufficient conditions for the asymptotic stability of the closed-loop system are obtained. Furthermore, under the setting of scalar-type local feedback gains, a method for constructing the input selection matrix is proposed. Numerical simulations show that the proposed method can achieve asymptotic stability of the closed-loop system by controlling only partial state components of a subset of nodes, and its feasibility is verified on larger-scale matrix-weighted networks. The results extend the iSCC-based stabilizability conclusions for scalar-weighted networks to the matrix-weighted case, and provide a theoretical reference for the decentralized stabilization design of matrix-weighted multi-agent systems.