Inertia theorems for matrices, controllability, and linear vibrations

Abstract

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 [5]. 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 [8] and [4] on a damping problem of the equation M x ̈ + (D + G) xdot; + Kx = 0 is extended. © 1974.