  In this paper, we continue investigating the second variation of Perelman's
$\nu$-entropy for compact shrinking Ricci solitons. In particular, we improve
some of our previous work in "H.-D. Cao and M. Zhu, Math. Ann. 353 (2012), No.
3, 747-763", as well as the more recent work in "M. Mehrmohamadi and A. Razavi,
arXiv:2104.08343", and obtain a necessary and sufficient condition for a
compact shrinking Ricci soliton to be linearly stable. Our work also extends
similar results of Hamilton, Ilmanen and the first author in
"arXiv:math.DG/0404165" (see also "H.-D. Cao and C. He, J. Reine Angew. Math.
2015 (2015), no. 709, 229-246.") for positive Einstein manifolds to the compact
shrinking Ricci soliton case.
