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} + 2rho 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).
matrix-decomposition algebraic-graph-theory spectral-graph-theory
asked Dec 1 at 4:11
Nandan Sriranga
11
add a comment |
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} + 2rho 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).
matrix-decomposition algebraic-graph-theory spectral-graph-theory
asked Dec 1 at 4:11
Nandan Sriranga
11
add a comment |
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} + 2rho 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).
matrix-decomposition algebraic-graph-theory spectral-graph-theory
asked Dec 1 at 4:11
Nandan Sriranga
11
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} + 2rho 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).
matrix-decomposition algebraic-graph-theory spectral-graph-theory
matrix-decomposition algebraic-graph-theory spectral-graph-theory
asked Dec 1 at 4:11
Nandan Sriranga
11
asked Dec 1 at 4:11
Nandan Sriranga
11
asked Dec 1 at 4:11
Nandan Sriranga
11
asked Dec 1 at 4:11
Nandan Sriranga
11
asked Dec 1 at 4:11
Nandan Sriranga
11
11
add a comment |
add a comment |
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