How to calculate a vec-permutation matrix? I'm trying to follow the maths in a paper on modelling population dynamics and have come unstuck on the calculation of the vec-permutation matrix. 
Let $p$ be the number of patches and $s$ be the number of stages. For my example, both $s$ and $p$ are $= 2$. 
The vec-permutation matrix has dimension $(sp × sp)$ and is given by:
$P(s,p) = \sum_{i=1}^{s}\sum_{j=1}^{p}E_{ij}\otimes E_{ij}^T$
The authors state $E_{ij}$ is an $s × p$ matrix with a $1$ in the $(i, j)$
position and zeros elsewhere and $\otimes$ denotes the Kronecker
matrix product.
From the paper, it should be the $4$ x $4$ matrix:
$P(2,2)= \begin{pmatrix}
1  & 0 & 0 & 0\\ 
0 & 0 & 1 & 0\\ 
0 & 1 & 0 & 0\\ 
0 & 0 & 0 & 1
\end{pmatrix} $
I don't understand the $E_{ij}$ matrix. I thought every matrix position could be considered $ij$. 
https://www.researchgate.net/profile/Christine_Hunter5/publication/228754096_The_use_of_the_vec-permutation_matrix_in_spatial_matrix_population_models/links/5551497808ae956a5d25ed48.pdf

Maybe looking at the $3 \times 3$ case will be helpful.  Note that
$$
\sum_{i,j} E_{ij} \otimes E_{ij} = 
\left[\begin{array}{ccc|ccc|ccc}
1&0&0&0&1&0&0&0&1\\
0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0\\
\hline
0&0&0&0&0&0&0&0&0\\
1&0&0&0&1&0&0&0&1\\
0&0&0&0&0&0&0&0&0\\
\hline
0&0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0&0\\
1&0&0&0&1&0&0&0&1
\end{array}
\right]
$$
If we take the transpose of each $3 \times 3$ block, we get the desired matrix
$$
\sum_{i,j} E_{ij} \otimes E_{ij}^T = 
\left[\begin{array}{ccc|ccc|ccc}
1&0&0&0&0&0&0&0&0\\
0&0&0&1&0&0&0&0&0\\
0&0&0&0&0&0&1&0&0\\
\hline
0&1&0&0&0&0&0&0&0\\
0&0&0&0&1&0&0&0&0\\
0&0&0&0&0&0&0&1&0\\
\hline
0&0&1&0&0&0&0&0&0\\
0&0&0&0&0&1&0&0&0\\
0&0&0&0&0&0&0&0&1
\end{array}
\right]
$$
