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Given a matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, its transpose, obviously, is $A^T=\begin{pmatrix}a&c\\b&d\end{pmatrix}$. But is there a conventional way of notating the matrix $\begin{pmatrix}d&b\\c&a\end{pmatrix}$ in terms of $A$? This generalizes in an obvious way to larger matrices. Also, do you know of some instance where this has been useful, especially in a geometric application?

I realize this seems like a dumb question, but I'm working on something where this matrix operation has come up in a significant way. It would be nice to see if anyone has had something similar happen.

For now I've been using $A_T$.

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    $\begingroup$ Without introducing new notations, this is $PA^TP$, with the matrix $P=[[0,1],[1,0]]$ . $\endgroup$
    – Pietro Majer
    Commented Jan 28, 2015 at 3:45
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    $\begingroup$ Yes that is clear but what I really want is something more concise. My use of it has it happening so regularly, I wouldn't want to write it out as a matrix product every time. $\endgroup$
    – j0equ1nn
    Commented Jan 30, 2015 at 3:39
  • $\begingroup$ Was calculating matrix-matrix exponentials for $M = [0, 2; 5, 0]$ and $N = [0, 5; 2, 0]$, and noticed that the left exponential ${}^N\!M$ and the right exponential $M^N$ are two similarly looking complex-valued matrices, which are the off-diagonal transposes of one another, thus came here to look for an appropriate notation. ${}^N\!M = (M^N)^t$ (with a small $t$ instead) should probably be fine. $\endgroup$
    – Buckminster
    Commented May 11 at 14:59

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In http://arxiv.org/abs/math/0701936 (Fuchsian equations of type DN, by Vasily Golyshev and Jan Stienstra) the transpose of the matrix $A$ with respect to the anti-diagonal is denoted by $A^\tau$. It relates to the ordinary transpose $A^T$ (or $A^t$ as used in the paper), as follows: $$A^\tau=JA^TJ$$ where $J=(J_{ij})_{0\le i,j\le n}$ denotes the matrix with $J_{ij}=1$ if $i+j=n$ and $J_{ij}=0$ otherwise. This fact was already noted by Pietro Majer for the case $n=1$ with notation $P$ instead of $J$ used in the Golyshev and Stienstra paper.

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    $\begingroup$ This is very nice. I also like the fact that they use a $J$. I don't know if it's a coincidence, but under the standard matrix representation $\rho$ of a quaternion algebra, the quaternion usually denoted by $j$ provides $\rho(jqj)^t=\rho(q)^{\tau}$ for any quaternion $q$. There's a hint of how I came to find a use for this by the way. $\endgroup$
    – j0equ1nn
    Commented Jan 30, 2015 at 3:44
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Since the transpose is defined around the diagonal and, to the best of my knowledge, the other diagonal is dubbed 'anti-diagonal', I think a natural extension would be call it 'anti-transpose' and write it $A^{aT}$.

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  • $\begingroup$ I would think a good way of writing it would be $A^{T'}$, but I don't know if that is used for some other purpose. $\endgroup$
    – Dennis Estenson
    Commented Apr 28, 2022 at 0:50
  • $\begingroup$ If $T$ denotes time ordering, then usually I see $\bar{T}$ being used for anti-time ordering. $\endgroup$
    – Bio
    Commented May 7 at 17:38

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