An inertia theorem for Lyapunovʹs equation and the dimension of a controllability space Original Research Article

Let A be an n × n complex matrix with inertia In(A) = (π(A), theta(A), δ(A)), and let H be an n × n hermitian matrix with inertia In(A) = (π(H), theta(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) less-than-or-equals, slant n − l and theta(A) − theta(H) less-than-or-equals, slant n − l. It is our purpose to prove this conjecture.