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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

Irregardless
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adkane
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  • I don't understand what the question is. As you say, The authors state $E_{ij}$ is an $s × p$ matrix with a $1$ in the $(i, j)$ position and zeros elsewhere. So, we know what $E_{ij}$ is supposed to mean. What's the problem? – Ben Grossmann Aug 08 '17 at 18:28
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    So, in the $2 \times 2$ case for example, $$ E_{12} = \pmatrix{0&1\0&0} $$ – Ben Grossmann Aug 08 '17 at 18:30
  • Even with that definition, I don't understand what that matrix looks like. Like I said I thought the (i,j) position covers all of the elements of a matrix. – adkane Aug 08 '17 at 18:31
  • I have no idea what that sentence means: "I thought the (i,j) position covers all elements of a matrix". – Ben Grossmann Aug 08 '17 at 18:39

1 Answers1

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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 $$ \sum_{i,j} E_{ij} \otimes E_{ij}^T = \left[\begin{array}{ccc|ccc|ccc} 1&0&0&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&1&0&0\\ \hline 0&1&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&0&0&0&1&0\\ \hline 0&0&1&0&0&0&0&0&0\\ 0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&0&1 \end{array} \right] $$

Ben Grossmann
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