The Klimushchev-Krasovskii theorem describes a sufficient condition for the asymptotic stability of singularly perturbed systems. Motivated by some subtle examples, this note presents a new necessary and sufficient condition by imposing an additional assumption on the statement of the Klimushchev-Krasovskii theorem. This note also extends the main inertia theorem to singularly perturbed systems. (C) 2011 Elsevier Ltd. All rights reserved.