  An extended time-dependent Hartree-Fock theory, known as the time-dependent
density-matrix theory (TDDM), is solved as a time-independent eigenvalue
problem for low-lying $2^+$ states in $^{24}$O to understand the foundation of
the rather successful time-dependent approach. It is found that the calculated
strength distribution of the $2^+$ states has physically reasonable behavior
and that the strength function is practically positive definite though the
non-hermitian hamiltonian matrix obtained from TDDM does not guarantee it. A
relation to an extended RPA theory with hermiticity is also investigated. It is
found that the density-matrix formalism is a good approximation to the
hermitian extended RPA theory.
