An inertia theorem for Lyapunov's equation and the dimension of a controllability space

AbstractLet A be an n × n complex matrix with inertia In(A) = (π(A), ϑ(A), δ(A)), and let H be an n × n hermitian matrix with inertia In(A) = (π(H), ϑ(H), δ(H)). Let K be an n × n positive semidefinite matrix such that K = AH + HA∗. Suppose that l is the dimension of the controllability space of the pair (A, K). Lerer and Rodman conjectured that |π(A) − π(H)| ⩽ n − l and |ϑ(A) − ϑ(H)| ⩽ n − l. It is our purpose to prove this conjecture