We numerically investigate the mechanical properties of static packings of frictionless ellipsoidal particles in two and three dimensions over a range of aspect ratio and compression Δφ. While amorphous packings of spherical particles at jamming onset (Δφ=0) are isostatic and possess the minimum contact number z_{iso} required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming (z<z_{iso}) from naive counting arguments, which assume that all contacts give rise to linearly independent constraints on interparticle separations. To understand this behavior, we decompose the dynamical matrix M=H-S for static packings of ellipsoidal particles into two important components: the stiffness H and stress S matrices. We find that the stiffness matrix possesses 2N(z_{iso}-z) eigenmodes e[over ̂]_{0} with zero eigenvalues even at finite compression, where N is the number of particles. In addition, these modes e[over ̂]_{0} are nearly eigenvectors of the dynamical matrix with eigenvalues that scale as Δφ, and thus finite compression stabilizes packings of ellipsoidal particles. At jamming onset, the harmonic response of static packings of ellipsoidal particles vanishes, and the total potential energy scales as δ^{4} for perturbations by amplitude δ along these "quartic" modes, e[over ̂]_{0}. These findings illustrate the significant differences between static packings of spherical and ellipsoidal particles.