If H is a Hermitian matrix and W = AH + HA* is positive definite, then A has as many eigenvalues with positive (negative) real part as H has positive (negative) eigenvalues . Theorems of this type are known as inertia theorems. In this note the rank of the controllability matrix of A and W is used to derive a new inertia theorem. As an application, a result in  and  on a damping problem of the equation M x ̈ + (D + G) xdot; + Kx = 0 is extended. © 1974.
Wimmer, H. K. (1974). Inertia theorems for matrices, controllability, and linear vibrations. Linear Algebra and Its Applications, 8(4), 337–343. https://doi.org/10.1016/0024-3795(74)90060-3