  When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it
is especially important to locate not only the unstable eigenvalues (i.e.,
those with positive real part), but also those which are purely imaginary but
have negative Krein signature. These latter eigenvalues have the property that
they can become unstable upon collision with other purely imaginary
eigenvalues, i.e., they are a necessary building block in the mechanism leading
to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a
general theory for constructing a meromorphic matrix-valued function, the
so-called Krein matrix, which has the property of not only locating the
unstable eigenvalues, but also those with negative Krein signature. These
eigenvalues are realized as zeros of the determinant. The resulting finite
dimensional problem obtained by setting the determinant of the Krein matrix to
zero presents a valuable simplification. In this paper the usefulness of the
technique is illustrated through prototypical examples of spectral analysis of
states that have arisen in recent experimental and theoretical studies of
atomic Bose-Einstein condensates. In particular, we consider one-dimensional
settings (the cigar trap) possessing real-valued multi-dark-soliton solutions,
and two-dimensional settings (the pancake trap) admitting complex multi-vortex
stationary waveforms.
