# Maximizing a certain eigenvalue ratio

Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following Lyapunov equation $$AP+PA^\top = -Q,$$ where $Q\in\mathbb{R}^{n\times n}$ is a positive definite matrix. Moreover, let $\lambda_{\min}(X)$ and $\lambda_{\max}(X)$ denote the smallest and largest (resp.) eigenvalue of the positive definite matrix $X$.

My question. For which positive definite $Q$ the following ratio $$r(Q) = \frac{\lambda_\min(Q)}{\lambda_\max(P)}$$ is maximized and what is the value of the maximum?