Notation fo the reverse identity matrix

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?

• What does this have to do with differential geometry? – JMoravitz Feb 14 '16 at 21:13
• that being said, en.wikipedia.org/wiki/Exchange_matrix uses the notation $J_n$ as well and cites Horn's Matrix Analysis. – JMoravitz Feb 14 '16 at 21:21
• 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. – user1551 Feb 14 '16 at 21:58

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.