Mathematically speaking, the eigenvalues of a square matrix A are the roots of its characteristic polynomial det(A − λI). An invariant subspace is a linear subspace that stays invariant under the action of A. In realistic applications, it usually takes a long process of simplifications, linearizations and discretiza- tions before one comes up with the problem of computing the eigenvalues of a matrix. In some cases, the eigenvalues have an intrinsic meaning, e.g., for the expected long-time behavior of a dynamical system; in others they are just meaningless intermediate values of a computational method. The same applies to invariant subspaces, which for example can describe sets of initial states for which a dynamical system produces exponentially decaying states.