Let $\mathbf{A}$ and $\mathbf{B}$ be real non-negative square matrices of the same size and $\mathbf{C}=\mathbf{A}+\mathbf{B}$.
Now consider the infinite block matrix $\mathbb{M}$ with $\mathbf{A}$ on the diagonal and $\mathbf{B}$ on the sub-diagonal.
Let us have $\mathbb{S}$ a symmetric matrix (of infinite size) and finally $\mathbb{K}=\mathbb{M}\mathbb{S}\mathbb{M}^{T}$ ($^T$ denotes transposition).
I know that $\mathbb{K}$ is symmetrical too. Now I consider $\mathbb{K}^{(t)}=\mathbb{M}^{t}\mathbb{S}{\big(\mathbb{M}^{t}}^{T}\big)$.
What can be said about the asymptotic behavior of $\mathbb{K}^{(t)}$ as $t \to \infty$?
Intuitively, I understand that $\mathbb{M}$ behaves in some ways like $\mathbf{C}$ as each block- column of the former sums to $\mathbf{A}+\mathbf{B}=\mathbf{C}$. However this does not seem to help understand the behavior of $\mathbb{M}^{t}$ from the maximum eigenvalue of $\mathbf{C}$ and related eigenvectors (or at least I failed to make the connection). Thanks a lot in advance