  This letter examines the controllability of consensus dynamics on
matrix-weighed networks from a graph-theoretic perspective. Unlike the
scalar-weighted networks, the rank of weight matrix introduces additional
intricacies into characterizing the dimension of controllable subspace for such
networks. Specifically, we investigate how the definiteness of weight matrices
influences the dimension of the controllable subspace. In this direction,
graph-theoretic characterizations of the lower and upper bounds on the
dimension of the controllable subspace are provided by employing, respectively,
distance partition and almost equitable partition of matrix-weighted networks.
Furthermore, the structure of an uncontrollable input for such networks is
examined. Examples are then provided to demonstrate the theoretical results.
