Suppose that $A$ is an asymmetric matrix that has all eigenvalues inside the unit circle. Let $Q$ be a symmetric, positive semidefinite matrix.
Let $W=\sum_{t=0}^{\infty} (A’)^t Q A^t$ a discrete time observability Gramian, or the solution to the discrete Lyapunov equation $A'WA-W+Q=0$.
If we know the spectrum of $A$ and $Q$, can we say anything about the spectrum of $W$?