ABSTRACT
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 x n +1 = A x n . 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 d − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429138492/cc3be78c-f644-49c2-b40d-db06d778c1a5/content/eq7947.tif"/> in ℝ d , 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 d with a fixed point y 0 ∈ ℝ d , can be described locally around y 0 via the linear differential equation ẋ = Dy f (y 0)x.
