The consensus problem over scalar-weighted graphs has attracted a lot of research attention from the control system and related engineering societies. The consensus algorithm provides simplified mathematical explanations for the collective behaviors observed in Reynolds’s boids and Vicsek’s models as well as laid a foundation for new research fields such as control of multiagent systems, networked control systems, etc. Many applications, such as formation control, network localization, distributed optimization and computation, network synchronization, resource allocation, etc., have been then proposed. It is also noteworthy that consensus-related models have been studied by diverse fields such as circuit theory, social networks, or applied mathematics (e.g., the cyclic pursuit problem). This chapter is devoted to the matrix-weighted consensus algorithm, a multi-dimensional extension of the widely studied consensus algorithm. A distinct feature of matrix-weighted consensus is the existence of clustering phenomena even if the topological graph is connected. Additional properties of the matrix-weighted consensus algorithms, which have been surveyed in the first chapter, will now be proved by rigorous mathematical arguments. First, some basic definitions regarding the matrix-weighted consensus problem will be provided. Second, the behaviors of the network under the matrix-weighted consensus algorithm will be examined under different assumptions of the matrix-weighted graphs. Third, the matrix-weighted consensus problem for double-integrator agents will be studied for directed matrix-weighted graphs (generally balanced and directed cycle). Finally, the matrix-weighted consensus algorithm will be designed for a network of higher-order integrators.