In this paper, linear vibrating systems, in which the inertia and stiffness matrices are symmetric positive definite and the damping matrix is symmetric positive semi-definite, are studied. Such a system may possess undamped modes, in which case the system is said to have residual motion. Several formulae for the number of independent undamped modes, associated with purely imaginary eigenvalues of the system, are derived. The main results formulated for symmetric systems are then generalized to asymmetric and symmetrizable systems. Several examples are used to illustrate the validity and application of the present results.