TITLE: What is the standard notation for reversing the order of vector's components?
QUESTION [8 upvotes]: If we have a vector $x=(x_1,x_2,\ldots,x_n)$, is there any standard way to denote the vector $(x_n,x_{n-1},\ldots,x_1)$?.
I think that $x^{-1}$ could be a good option.

REPLY [10 votes]: An alternative would be to define and use the exchange matrix (see the Wikipedia entry “Exchange matrix”)
$$
  J = \begin{pmatrix}
0 & 0 &\cdots &0 & 1\\
0 & 0 & \cdots & 1 & 0\\
\vdots & \vdots &\ddots & 0 & 0\\
0 & 1 &\cdots & 0 &0\\
1 & 0 &\cdots & 0 &0
\end{pmatrix}
$$
and to note that $(x_n, \dotsc, x_1)=J (x_1, \dotsc, x_n)$.