On the Kalman–Yacubovich–Popov lemma and common Lyapunov solutions for matrices with regular inertia

In this paper we extend the classical Lefschetz version of the Kalman–Yacubovich–Popov (KYP) lemma to the case of matrices with general regular inertia. We then use this result to derive an easily verifiable spectral condition for a pair of matrices with the same regular inertia to have a common Lyapunov solution (CLS), extending a recent result on CLS existence for pairs of Hurwitz matrices.