Dynamical Systems and Linear Algebra

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 ẋ = Ax, 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 smooth manifolds, such as the sphere S in R, the Grassmann manifolds, or on classical (matrix) Lie groups. Again, the behavior of such systems is closely related to matrices and their properties. And the behavior of nonlinear systems, e.g. of differential equations ẏ = f(y) in R with a fixed point y0 ∈ R d can be described locally around y0 via the linear differential equation ẋ = Dyf(y0)x. Since A.M. Lyapunov’s thesis in 1892 it has been an intriguing problem how to construct an appropriate linear algebra for time varying systems. Note that, e.g., for stability of the solutions of ẋ = A(t)x it is not sufficient that for all t ∈ R the matrices A(t) have only eigenvalues with negative real part (see [Hah67], Chapter 62). Of course, Floquet theory (see [Flo83]) gives an elegant solution for the periodic case, but it is not immediately clear how to build a linear algebra around Lyapunov’s ‘order numbers’ (now called Lyapunov exponents). The multiplicative ergodic theorem of Oseledets [Ose68] resolves the issue for measurable linear systems with stationary time dependencies, and the Morse spectrum together with Selgrade’s theorem [Sel75] clarifies the situation for continuous linear systems with chain transitive time dependencies. This section provides a first introduction to the interplay between linear algebra and analysis/topology in continuous time. Subsection 1 recalls facts about d-dimensional linear differential