On the Lyapunov equation with the state matrix in companion form (opens in new tab)

We study the continuous-time Lyapunov equation under the assumption that the state matrix is a Hurwitz companion matrix. The standard Lyapunov theory implies that the unique solution $X$ is positive semidefinite. Motivated by positive systems, we investigate the question of whether $X$ is entrywise nonnegative. We prove that this is the case when the companion matrix has only real eigenvalues. The proof reduces each entry of $X$ to a quadratic f...