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What would be the matrix form of $P$ defined in equation (1.11) of this published work, where (I have chosen $n=2$, for simplicity)

$$P_{kl}=\begin{cases} \delta_{k,2l-1} ~~~ k \le 2\\\\ \delta_{2+k,2l} ~~~ l\le 2 \end{cases}$$

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  • $\begingroup$ Try computing the components??? $\endgroup$
    – K.defaoite
    Dec 13, 2021 at 7:56
  • $\begingroup$ @K.defaoite what happens when $k, l > 2$ or $k, l < 2$. Eg what's $P_{11}$ or $P_{34}$? $\endgroup$
    – okzoomer
    Dec 13, 2021 at 8:19
  • $\begingroup$ @okzoomer, that's exactly my point! $\endgroup$
    – User101
    Dec 14, 2021 at 9:49

1 Answer 1

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From the context of the paper (namely the text below Equation (1.2) that defines $\mathbf R$ and the text below Equation (1.4) defining $\mathbf S$), it seems that the permutation matrix $\mathbf P$ being discussed for which $\mathbf P \mathbf R = \mathbf S$ is the commutation matrix $\mathbf P = \mathbf K^{(2,n)}$.

Some examples:

$$n = 2: \mathbf P = \left[\begin{array}{cc|cc}1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\ \hline0 & 1 & 0 & 0\\0 & 0 & 0 & 1\end{array}\right]\\ n=3: \mathbf P = \left[\begin{array}{cc|cc|cc} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ \hline 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]\\ n=4: \mathbf P = \left[\begin{array}{cc|cc|cc|cc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]. $$ With that, I suspect that the intended description of $\mathbf P$ was as follows. $$ P_{kl} = \begin{cases} \delta_{k,2l-1} & k \leq n,\\ \delta_{k-n,2l-2} & k > n. \end{cases} $$

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