Pseudospectra for exponential polynomial matrices

ẋ(t) = Ax(t) (1) where x ∈ C and A ∈ C(n×n), together with (say) initial conditions x(0) = x0, occurs often as a simple model of many applied dynamical phenomena, for instance in theoretical evolution or in the physics of lasers, to name only two out of many possibilities. The so-called characteristic equation det(λI − A) = 0 occurs when looking for solutions of the form x(t) = ev for some vector v ∈ C. Understanding the exact solution x(t) = exp(At)x0 comes from the eigenvalues (spectrum) of A, and more recently from the pseudospectrum of A, by which is meant the set [5, 17]