Characterizing bipartite consensus on signed matrix-weighted networks via balancing set

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