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In the Wikipedia page for the Pauli matrices, there is a list of the eigenvectors of the Pauli matrices.

Notice at $\sigma_y$, it's eigenvectors are $\begin{pmatrix} 1 \\ \pm i \end{pmatrix}$ but not $\begin{pmatrix} \pm i\\ 1 \end{pmatrix}$, same for $\sigma_x$ and $\sigma_y$.

By which formalism or customs stated that Pauli matrices' eigenvectors are $\begin{pmatrix} 1 \\ \pm i \end{pmatrix}$ but not $\begin{pmatrix} \pm i\\ 1 \end{pmatrix}$?

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    $\begingroup$ They are equal, modulo an $i$ or two. People liked the 1 up top... $\endgroup$ – Jon Custer Sep 3 '18 at 13:58
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The standard way to encode vectors on the Bloch sphere sets the first component as a positive real number, and this convention is influenced by that.

But that's about it: it is a convention, and you're welcome to change it if there's a specific reason why that would suit your purposes better.

However, if there isn't a well-defined reason why it's better for you in some measure, then I would advice against trying to buck this convention: the only thing that it's going to achieve is that a good fraction of your readers is going to waste time wondering why the phase convention on your eigenbases has been chosen in a non-standard way, and you want your readers' attention to be focused on what you want to say instead of on your notation. The positive-first-component is just a part of the shared language that makes it easier to communicate clearly about the content; there's no good reason to make one choice over others (unlike in other domains) but having a common choice that breaks that degeneracy takes that uncertainty out and leaves more room for the physics.

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