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I'm wondering if there's a canonical notation for the reverse identity matrix, i.e. $$ ?=\left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 1\\ 0& 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0& 0 & 0\\ 1&0 & 0 & 0 & 0 \end{array}\right) $$ Sometimes I've seen $J_n$ but I'm not quite satysfied since I usually use this notation for the antisymmetric version of this matrix. Is it a standard notation?

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    $\begingroup$ What does this have to do with differential geometry? $\endgroup$ – JMoravitz Feb 14 '16 at 21:13
  • $\begingroup$ that being said, en.wikipedia.org/wiki/Exchange_matrix uses the notation $J_n$ as well and cites Horn's Matrix Analysis. $\endgroup$ – JMoravitz Feb 14 '16 at 21:21
  • $\begingroup$ While $J_n$ is a relatively popular notation for the reversal matrix, I don't think it's a standard notation. Many people use $J_n$ to denote something else, such as an $n\times n$ Jordan block or the all-one matrix. $\endgroup$ – user1551 Feb 14 '16 at 21:58
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I don't think that there is a widely used symbol for that. I would consider the permutation $$ \overset{\tiny \leftarrow}{\sigma}:=\begin{pmatrix} n & n-1 & \ldots &2 & 1 \end{pmatrix}$$ and then refer to that matrix as $P_\overset{\tiny \leftarrow}{\sigma}$. You can of course drop the small arrow if it seems too heavy, or use another (Greek) letter for the permutation.

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