Matrix Theory for Cooperative Systems

Abstract

In this chapter, non-negative matrices and their properties are introduced. Among the important concepts to characterize non-negative matrices are irreducibility and reducibility, cyclicity and primitiveness, and lower-triangular completeness. The so-called Perron-Frobenius theorem provides the fundamental results on eigenvalues and eigenvectors of a non-negative matrix. Non-negative matrices arise naturally from systems theory, in particular, non-negative matrices for discrete-time positive systems and/or cooperative systems, row-stochastic matrices for Markov chains and cooperative systems, Metzler matrices for continuous-time positive systems and/or cooperative systems, and M-matrices for asymptotically stable systems. Geometrical, graphical and physical meanings of non-negative matrices are explicitly shown. Analysis tools and useful properties of the aforementioned matrices are detailed for their implications in and applications to dynamical systems, especially cooperative systems. Specifically, eigenvalue properties are explored, existence of and explicit solution to Lyapunov function are determined, and convergence conditions on matrix sequences are obtained.