Inertia theorems for pairs of matrices

Abstract

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 this paper, we generalize Lyapunov’s theorem by giving necessary and sufficient conditions for ( A, B) being positive stabilizable. We also give generalizations of the main inertia theorem and of another inertia theorem due to Chen and Wimmer. Then we deduce a necessary condition for the existence of a Hermitian matrix H such that K:= LH+ HL * is positive semidefinite and the number of nonconstant invariant factors of xI−LK has a fixed value. This last result was inspired by another inertia theorem due to Loewy.