  In this paper we consider gauge invariant linear perturbations of the metric
and matter tensors describing the self-similar Lema\^itre-Tolman-Bondi
(timelike dust) spacetime containing a naked singularity. We decompose the
angular part of the perturbation in terms of spherical harmonics and perform a
Mellin transform to reduce the perturbation equations to a set of ordinary
differential equations with singular points. We fix initial data so the
perturbation is finite on the axis and the past null cone of the singularity,
and follow the perturbation modes up to the Cauchy horizon. There we argue that
certain scalars formed from the modes of the perturbation remain finite,
indicating linear stability of the Cauchy horizon.
