  A matrix method is formulated in a Lagrangian representation for the solution
of the characteristic value problem governing modes of oscillation and
instability in a collisionless stellar system. The underlying perturbation
equations govern the Lagrangian displacement of a star in the six-dimensional
phase space. This matrix method has a basis in a variational principle. The
method is developed in detail for radial oscillations of a spherical system.
The basis vectors required for the representation of the Lagrangian
displacement in this case are derived from solutions of the Lagrangian
perturbation equations for radial perturbations in a homogeneous sphere. The
basis vectors are made divergence free in the six-dimensional phase space in
accordance with the requirement of Liouville's theorem that the flow of the
system in the phase space must be incompressible. The basis vectors are made
orthogonal with respect to a properly chosen set of adjoint vectors with the
aid of a Gram-Schmidt procedure. Some basis vectors are null vectors in the
sense that their inner products with their own adjoint vectors vanish. The
characteristic frequencies of the lowest radial modes are calculated in several
approximations for members of a family of spherical models which span a wide
range of central concentrations. The present formulation of the matrix method
can be generalized for nonradial modes in spherical systems and for modes in
axisymmetric systems.
