Inertia theorems for pairs of matrices Original Research Article

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 ofimage has a fixed value. This last result was inspired by another inertia theorem due to Loewy.