Consensus of Second-order Matrix-weighted Multi-agent Networks

This paper investigates consensus problem of second-order multi-agent system on matrix-weighted networks. It is shown that when the null space of the Gauge transformed graph Laplacian is spanned by the Kronecker product of an all-one vector and a set of orthogonal vectors, the algebraic multiplicity of eigenvalue zero cannot exceed the nullity of the graph Laplacian, thus admitting a proper blocking of the system matrix's Jordan normal form. Second-order bipartite consensus is thereby achieved independent of the structural balance of the network. Simulation examples are provided to demonstrate the theoretical results.