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).


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