Handbook of Linear Algebra, 2nd Edition

Chapter 79

Dynamical Systems and Linear Algebra

Fritz Colonius

Universität Augsburg

Wolfgang Kliemann

Iowa State University

Linear algebra plays a key role in the theory of dynamical systems, and concepts from dynamical systems allow the study, characterization, and generalization of many objects in linear algebra, such as similarity of matrices, eigenvalues, and (generalized) eigenspaces. The most basic form of this interplay can be seen as a matrix A gives rise to a continuous time dynamical system via the linear ordinary differential equation $\dot{\text{x}}=A\text{x}$, or a discrete time dynamical system via iteration xn+1 = Axn. The properties of the solutions are intimately related to the properties of the matrix A. Matrices also define nonlinear systems on ...