In the following, let $P$ be a stochastic matrix, i.e. $\sum_{j} P_{ij} = 1, P_{ij} \geq 0.$ Assume furthermore $P$ is irreducible and aperiodic, which implies there is a unique stationary distribution $\pi$ (here a vector) such that $\pi P = P$. Let $\Pi$ be the diagonal matrix with $\pi$ on the diagonal. Furthermore, let $\gamma \in [0,1)$.
Now, consider the linear map
$$Q(U) = \left(Id-\gamma P^T\right)^{-1}U\Pi \left(Id-\gamma P\right)^{-1},$$
which maps $$\mathbb{R}^{n \times n} \to \mathbb{R}^{n \times n}.$$
I would like to know about the eigenvalues of this linear map, especifically about the sign of the real parts - in particular whether they are always non-negative.
Here are some considerations:
- For matrices $A,B$, we have $$AUB = (A \otimes B^T) \operatorname{vec}(U),$$ where $\otimes$ denotes the Kronecker product.
- The eigenvalues of the Kronecker-Product $$A \otimes B^T$$ are all of the form $\lambda_i \mu_j$, where $\lambda_i$ is an eigenvalue of $A$ and $\mu_j$ is an eigenvalue of $B$
- We can rewrite $$Q(U) = \left(Id-\gamma P^T\right)^{-1}U\left(Id-\gamma P^T\right)^{-1}\left(Id-\gamma P^T\right)\Pi \left(Id-\gamma P\right)^{-1}$$
- The matrix $\left(Id-\gamma P^T\right)\Pi$ has eigenvalues with all positive real parts, and hence also has $(Id-\gamma P^T)^{-1}\left(Id-\gamma P^T\right)\Pi \left(Id-\gamma P\right)^{-1}$.
That's about all I know, any ideas how to proceed?