Convergence of Adjacency Matrix: Spectral Properties and Consensus Theorem

This lecture covers the convergence of the powers of the adjacency matrix and the consensus theorem for primitive and stochastic matrices. It delves into the spectral properties of non-negative matrices, including the Perron-Frobenius theorem, which characterizes the eigenvalues and eigenvectors. The lecture also discusses the convergence of networked control systems, emphasizing the importance of primitivity and stochasticity in achieving consensus. Examples in wireless sensor networks are used to illustrate the concepts, highlighting the relationship between matrix properties and connectivity in digraphs. The rate of convergence to consensus and the essential spectral radius are explored, providing insights into the exponential convergence rate. Key takeaways include the significance of non-negative, irreducible, and primitive matrices in networked systems.