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For even $n = 2k$, you can take a permutation matrix belonging to $k$ disjoint transpositions like $(12)(34)\ldots(n-1,n)$. This is self-inverse, so in particular similar to its inverse.
| | 1 | initial version |
For even $n = 2k$, you can take a permutation matrix belonging to $k$ disjoint transpositions like $(12)(34)\ldots(n-1,n)$. This is self-inverse, so in particular similar to its inverse.
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