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.
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1$\begingroup$ For a permutation of indices $\sigma$, I would denote $x_\sigma:=(x_{\sigma_1},\dots x_{\sigma_n})$, (or $x\sigma$ or $x\circ\sigma$ or $x^\sigma$). $\endgroup$– Pietro MajerCommented Dec 9, 2016 at 11:26
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1$\begingroup$ @PietroMajer: And how do you denote the permutation $\sigma(i)=n+1-i$ of $\{1,\dotsc,n\}$? $\endgroup$– HeinrichDCommented Dec 9, 2016 at 11:46
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2$\begingroup$ My 2c: how about $x^\leftarrow$? $\endgroup$– Anthony QuasCommented Dec 9, 2016 at 11:53
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$\begingroup$ @ HeinrichD, yes, this is the question $\endgroup$– Pietro MajerCommented Dec 9, 2016 at 12:04
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2$\begingroup$ Instead of $x$, name your vector $b$ and the reversed one $d$. $\endgroup$– monkeymathsCommented Dec 9, 2016 at 12:33
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2 Answers
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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 $(x_n, \dotsc, x_1)=J (x_1, \dotsc, x_n)$.
answered Dec 9, 2016 at 11:56
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I think that $\mathrm{flip}(x)$ is a better choice. This is also used in some programming languages. Notice that $x^{-1}$ could be easily confused with $(x_1^{-1},\dotsc,x_n^{-1})$.
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$\begingroup$ Hm, actually, just $\mathrm{reverse}$ seems to be more common. See e.g. cplusplus.com/reference/list/list/reverse $\endgroup$ Commented Dec 9, 2016 at 11:44