  In this paper we provide a detailed proof of the second variation formula,
essentially due to Richard Hamilton, Tom Ilmanen and the first author, for
Perelman's $\nu$-entropy. In particular, we correct an error in the stability
operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary
condition for linearly stable shrinkers in terms of the least eigenvalue and
its multiplicity of certain Lichnerowicz type operator associated to the second
variation.
