  Using the Jacobian matrix, we obtain theoretical expression of rigidity and
the density of states of two-dimensional amorphous solids consisting of
frictional grains in the linear response to an infinitesimal strain, in which
we ignore the dynamical friction caused by the slip processes of contact
points. The theoretical rigidity agrees with that obtained by molecular
dynamics simulations. We confirm that the rigidity is smoothly connected to the
value in the frictionless limit. For the density of states, we find that there
are two modes in the density of states for sufficiently small $k_{T}/k_{N}$,
which is the ratio of the tangential to normal stiffness. Rotational modes
exist at low frequencies or small eigenvalues, whereas translational modes
exist at high frequencies or large eigenvalues. The location of the rotational
band shifts to the high-frequency region with an increase in $k_{T}/k_{N}$ and
becomes indistinguishable from the translational band for large $k_{T}/k_{N}$.
The rigidity determined by the translational modes agrees with that obtained by
the molecular dynamics simulations, whereas the contribution of the rotational
modes is almost zero for small $k_{T}/k_{N}$.
