  I show that the kernel of the random phase approximation (RPA) matrix based
on a stable Hartree, Hartree-Fock, Hartree-Bogolyubov or
Hartree-Fock-Bogolyubov mean field solution is decomposed into a subspace with
a basis whose vectors are associated, in the equivalent formalism of a
classical Hamiltonian homogeneous of second degree in canonical coordinates,
with conjugate momenta of cyclic coordinates (Nambu-Goldstone modes) and a
subspace with a basis whose vectors are associated with pairs of a coordinate
and its conjugate momentum neither of which enters the Hamiltonian at all. In a
subspace complementary to the one spanned by all these coordinates including
the conjugate coordinates of the Nambu-Goldstone momenta, the RPA matrix
behaves as in the case of a zerodimensional kernel. This result was derived
very recently by Nakada as a corollary to a general analysis of RPA matrices
based on both stable and unstable mean field solutions. The present proof does
not rest on Nakada's general results.
