# Decomposition of a block matrix related to graph Laplacians

I'm working on graph networks in a communications problem and I have trouble understanding how to work with this $$3 \times 3$$ block matrix related to graph Laplacians:

D = $$\begin{bmatrix} \rho D_0 L_{+} & -D_0 & D_0 \\ \rho^{2}L_-D_0L_+ & \textbf{I}-\rho L_-D_0 & \rho L_-D_0 \\ \textbf{0} & \textbf{0} & \textbf{I} \end{bmatrix}$$

where, $$\begin{equation*} D_0 = (\textbf{I} + 2\rho W)^{-1} \end{equation*}$$

$$L_{+}$$ and $$L_{-}$$ are the unsigned and signed Laplacian matrices of the underlying graph and $$W$$ is the diagonal matrix with diagonal entry $$(i,i)$$ having the number of neighbours of node $$i$$. These matrices are also $$positive \thinspace semidefinite$$.

$$\rho$$ is just a constant.

The graphs are also $$balanced \thinspace digraphs$$, in the sense that if two nodes are connected, there are two edges between them; one from each node to the other.

All matrices embedded in the block matrix are $$n \times n$$ matrices.

What I aim to do are to understand the $$spectral$$ properties of $$D^k$$ through some decomposition theorems (This is what I think will help me the most and I would like some references regarding this). I have read about $$Schur \thinspace decomposition$$ but I could find literature only on $$2 \times 2$$ block matrices. More specifically, I would like to understand what happens to $$D^{\infty}$$(i.e its spectral properties).