6 displays an example with only one pair of complex conjugate eigenvalues and three real eigenvalues with two of them on the same side of the real part of the complex eigenvalues (Example 3).
We generally have more than 3 unknowns (except for n = 6 with three pairs of complex conjugate eigenvalues) and therefore an underdetermined linear system for the unknowns [[omega].
Therefore, we can apply the same elimination technique as before by considering the matrices of order 3 corresponding to all the triples of eigenvalues, a pair of complex conjugate ones counting only for one.
7 corresponds to an example with n = 6 and three pairs of complex conjugate eigenvalues (Example 4).
Things are strikingly different if we construct a real normal matrix with the given eigenvalues (which are real or occur in complex conjugate pairs) and run the Arnoldi method with real starting vectors.
Clearly we cannot do the same as for k = 2 because, for instance, for k = 3, we have either three real Ritz values or a pair of complex conjugate Ritz values and a real one.
3 shows the Ritz values at iteration 5 (that is, the next to last one); there is an accumulation of some Ritz values on this spurious curve as well as close to the other pair of complex conjugate eigenvalues.
In this paper we concentrated on pairs of complex conjugate Ritz values, but an interesting problem is to locate the real Ritz values in the intersection of the field of values with the real axis.
Furthermore, in the case where A(t) is constant but has complex conjugate eigenvalues, the averaging in (2.
There is much scope for further theoretical work in this area, including (a) fully analysing the case of complex conjugate eigenvalues and (b) extending the rigorous analysis to more general problem classes, such as the Floquet case [4, pages 412-413].