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We suppose the matrix $A\in \mathbb{R}^{2n\times 2n}$ is in the following form ‎\begin{eqnarray} ‎S=\left(‎ ‎\begin{array}{cc}‎ ‎S_1 & S_2 \\‎ ‎S_2 & S_1 \\‎ ‎\end{array}‎ ‎\right)‎, ‎\end{eqnarray}‎ where the matrices $S_1, S_2 \in \mathbb{R}^{n\times n}$. How we can apply the permutation matrices (i.e., by multiplying $S$ with the permutation matrices in the left and right sides of $S$) to create the new matrix $S^{'}$ in the following form \begin{eqnarray} ‎S^{'}=\left(‎ ‎\begin{array}{cc}‎ ‎S_1 & S_2 \\‎ ‎S_1 & S_2 \\‎ ‎\end{array}‎ ‎\right)‎. ‎\end{eqnarray}‎

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There can not be block permutation matrices $P$ and $Q$ such that $S’=PSQ$.

In fact, the only possibility is $$ P = Q = \begin{bmatrix} 0 & I_n \\ I_n & 0 \end{bmatrix}, $$ And this doesn’t work.

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