# Matrix written in terms of Kronecker delta

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)

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

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

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