  Riemannian manifolds with non-zero Killing spinors are Einstein manifolds.
Klaus Kr\"{o}ncke proved that all complete Riemannian manifolds with imaginary
Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper,
we obtain a new proof for this stability result by using a Bochner type formula
in \cite{DWW05} and \cite{Wan91}. Moreover, existence of real Killing spinors
is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein
manifold is essentially the total space of a certain principal $S^{1}$-bundle
over a K\"{a}hler-Einstein manifold. We prove that if the base space is a
product of two K\"{a}hler-Einstein manifolds then the regular Sasaki-Einstein
manifold is unstable. This provides us many new examples of unstable manifolds
with real Killing spinors.
