Inertia theorems for pairs of matrices, II

Let L be a square matrix. A well-known theorem due to Lyapunov states that L is positive stable if and only if there exists a (Hermitian) positive definite matrix H such that LH + HL∗ is positive definite. The main inertia theorem, due to Ostrowski, Schneider and Taussky, states that there exists a Hermitian matrix H such that LH + HL∗ is positive definite if and only if L has no eigenvalues with zero real part; and, in that case, the inertias of L and H coincide.A pair (A, B) of matrices of sizes p × p and p × q, respectively, is said to be positive stabilizable if there exists X such that A + BX is positive stable. In a previous paper, the results above and other inertia theorems were generalized to pairs of matrices, in order to study stabilization instead of stability. In a second paper, analogous questions about stabilization with respect to the unit disc were also considered.Denote by π(L) the number of eigenvalues of L with real positive part. In the present paper, we study the inequality π(LH + HL∗) ⩾ l, the corresponding inequality for discrete-time systems, π(H − LHL∗) ⩾ l, and their generalizations related with stabilization.