Grounded Laplacians of Directed Signed Matrix-Weighted Networks: Spectral Properties and Applications to Non-Trivial Consensus

Grounded Laplacians provide the spectral link between external information and network convergence. This paper establishes positive-stability results for grounded Laplacians in directed signed matrix-weighted networks, where directionality, antagonism, and singular edge weight matrices coexist. First, under in-degree dominance and positive-negative reachability, we derive explicit local thresholds for the grounding gains. Second, a scaled, kernel-based certificate replaces the unscaled degree condition with a signed matrix-weighted Dirichlet decomposition and a joint-kernel test for the scaled symmetric part. The computable margin $γ_p$ lower-bounds the minimum real part of the spectrum and certifies exponential contraction in the $P$-norm. Under absolute generalized balance, the kernel-intersection test is given; the balanced and definite-edge unbalanced undirected cases follow. As an application, non-trivial consensus (NTC) on signed matrix-weighted networks is studied. Informed agents, external signals and coupling terms are designed to steer all agents to any prescribed nonzero state without requiring structural balance. Switching topology case retains non-trivial consensus result under certain conditions. Realizing NTC on signed matrix-weighted networks demonstrates that groups with both cooperative and antagonistic multi-dimensional interactions can achieve consensus, which was previously deemed exclusive to fully cooperative groups.