Spectral Properties of Graph Matrices via Convergence Analysis of Linear Dynamical Systems

This paper studies the spectral properties of two important graph matrices in the literature, known as the leader-follower matrices. These two matrices arise from various cooperative control problems of multi-agent systems in a leader-follower architecture, and in most cases, they are the keys to the solvability of these cooperative control problems. In contrast with the existing results, which appeal to the algebraic graph theory and matrix analysis tools, we reveal the spectral properties from the viewpoint of dynamical systems. In particular, the spectral properties of the two graph matrices are derived by analyzing the convergence of a continuous-time and a discrete-time linear system, respectively.