Linear algebra plays a key role in the theory of dynamical systems, and
concepts from dynamical systems allow the study, characterization and gen-
eralization of many objects in linear algebra, such as similarity of matrices,
eigenvalues, and (generalized) eigenspaces. The most basic form of this in-
terplay can be seen as a quadratic matrix A gives rise to a discrete time
dynamical system xk+1 = Axk, k = 0, 1, 2,... and to a continuous time
dynamical system via the linear ordinary differential equation ˙ x = Ax.
The (real) Jordan form of the matrix A allows us to write the solution
of the differential equation ˙ x = Ax explicitly in terms of the matrix ex-
ponential, and hence the properties of the solutions are intimately related
to the properties of the matrix A. Vice versa, one can consider properties
of a linear flow in
and infer characteristics of the underlying matrix A.
Going one step further, matrices also define (nonlinear) systems on smooth
manifolds, such as the sphere Sd−1 in Rd, the Grassmannian manifolds, the
flag manifolds, or on classical (matrix) Lie groups. Again, the behavior of
such systems is closely related to matrices and their properties.
Since A.M. Lyapunov’s thesis  in 1892 it has been an intriguing prob-
lem how to construct an appropriate linear algebra for time-varying systems.
Note that, e.g., for stability of the solutions of ˙ x = A(t)x it is not suﬃcient
that for all t ∈ R the matrices A(t) have only eigenvalues with negative
real part (see, e.g., Hahn , Chapter 62). Classical Floquet theory (see
Floquet’s 1883 paper ) gives an elegant solution for the periodic case,
but it is not immediately clear how to build a linear algebra around Lya-
punov’s ‘order numbers’ (now called Lyapunov exponents) for more general
time dependencies. The key idea here is to write the time dependency as a