  In its original formulation the Krein matrix was used to locate the spectrum
of first-order star-even polynomial operators where both operator coefficients
are nonsingular. Such operators naturally arise when considering
first-order-in-time Hamiltonian PDEs. Herein the matrix is reformulated to
allow for operator coefficients with nontrivial kernel. Moreover, it is
extended to allow for the study of the spectral problem associated with
quadratic star-even operators, which arise when considering the spectral
problem associated with second-order-in-time Hamiltonian PDEs. In conjunction
with the Hamiltonian-Krein index (HKI) the Krein matrix is used to study two
problems: conditions leading to Hamiltonian-Hopf bifurcations for small
spatially periodic waves, and the location and Krein signature of small
eigenvalues associated with, e.g., $n$-pulse problems. For the first case we
consider in detail a first-order-in-time fifth-order KdV-like equation. In the
latter case we use a combination of Lin's method, the HKI, and the Krein matrix
to study the spectrum associated with $n$-pulses for a second-order-in-time
Hamiltonian system which is used to model the dynamics of a suspension bridge.
