  We show that a spacetime satisfying the linearized vacuum Einstein equations
around a type D background is generically of type I, and that the splittings of
the Principal Null Directions (PNDs) and of the degenerate eigenvalue of the
Weyl tensor are non analytic functions of the perturbation parameter of the
metric. This provides a gauge invariant characterization of the effect of the
perturbation on the underlying geometry, without appealing to differential
curvature invariants. This is of particular interest for the Schwarzschild
solution, for which there are no signatures of the even perturbations on the
algebraic curvature invariants. We also show that, unlike the general case, the
unstable even modes of the Schwarzschild naked singularity deforms the Weyl
tensor into a type II one.
