  We study the weighted average consensus problem for a gossip network of
agents with vector-valued states. For a given matrix-weighted graph, the gossip
process is described by a sequence of pairs of adjacent agents communicating
and updating their states based on the edge matrix weight. Our key contribution
is providing conditions for the convergence of this non-homogeneous Markov
process as well as the characterization of its limit set. To this end, we
introduce the notion of "$w$-holonomy" of a set of stochastic matrices, which
enables the characterization of sequences of gossiping pairs resulting in
reaching a desired consensus in a decentralized manner. Stated otherwise, our
result characterizes the limiting behavior of infinite products of
(non-commuting, possibly with absorbing states) stochastic matrices.
