Particle-Hole Matrix: Its Connection with the Symmetries and Collective Features of the Ground State

A detailed study is made of the properties of the ``particle-hole matrix''Gabcd≡〈g| (a†b−ρab)†(c†d−ρcd) |g〉,where |g> is a many-fermion ground state and ρab is the 1-body density matrix. It is shown that the zero eigenvalues of the particle-hole matrix are intimately and simply related to the one-body symmetries of the ground state. It is also shown that the large eigenvalues of Gabcd are closely related to the collective features of the ground state.