Introduction

Background

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

Rd

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 [97] 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 [61], Chapter 62). Classical Floquet theory (see

Floquet’s 1883 paper [50]) 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

xi