  This paper examines the consensus problem on time-varying matrix-weighed
undirected networks. First, we introduce the matrix-weighted integral network
for the analysis of such networks. Under mild assumptions on the switching
pattern of the time-varying network, necessary and/or sufficient conditions for
which average consensus can be achieved are then provided in terms of the null
space of matrix-valued Laplacian of the corresponding integral network. In
particular, for periodic matrix-weighted time-varying networks, necessary and
sufficient conditions for reaching average consensus is obtained from an
algebraic perspective. Moreover, we show that if the integral network with
period $T>0$ has a positive spanning tree over the time span $[0,T)$, average
consensus for the node states is achieved. Simulation results are provided to
demonstrate the theoretical analysis.
