  In contrast with the scalar-weighted networks, where bipartite consensus can
be achieved if and only if the underlying signed network is structurally
balanced, the structural balance property is no longer a graph-theoretic
equivalence to the bipartite consensus in the case of signed matrix-weighted
networks. To re-establish the relationship between the network structure and
the bipartite consensus solution, the non-trivial balancing set is introduced
which is a set of edges whose sign negation can transform a structurally
imbalanced network into a structurally balanced one and the weight matrices
associated with edges in this set have a non-trivial intersection of null
spaces. We show that necessary and/or sufficient conditions for bipartite
consensus on matrix-weighted networks can be characterized by the uniqueness of
the non-trivial balancing set, while the contribution of the associated
non-trivial intersection of null spaces to the steady-state of the
matrix-weighted network is examined. Moreover, for matrix-weighted networks
with a positive-negative spanning tree, necessary and sufficient condition for
bipartite consensus using the non-trivial balancing set is obtained. Simulation
examples are provided to demonstrate the theoretical results.
