Editing Pauli matrices (section)

====Adjoint action====
It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle <math>a</math> along any axis <math>\hat n</math>:
<math display=block>  
  R_n(-a) ~ \vec{\sigma} ~ R_n(a) =
  e^{i \frac{a}{2}\left(\hat{n} \cdot \vec{\sigma}\right)} ~ \vec{\sigma} ~  e^{-i \frac{a}{2}\left(\hat{n} \cdot \vec{\sigma}\right)} =
  \vec{\sigma}\cos (a) + \hat{n} \times \vec{\sigma} ~ \sin(a) + \hat{n} ~ \hat{n} \cdot \vec{\sigma} ~ (1 - \cos(a)) ~ .
</math>

Taking the dot product of any unit vector with the above formula generates the expression of any single qubit operator under any rotation. For example, it can be shown that <math display="inline">R_y\mathord\left(-\frac{\pi}{2}\right)\, \sigma_x\, R_y\mathord\left(\frac{\pi}{2}\right) = \hat{x} \cdot \left(\hat{y} \times \vec{\sigma}\right) = \sigma_z</math>.

{{see also|Rodrigues' rotation formula}}