Editing Pauli matrices (section)

=== Eigenvalues and eigenvectors ===

The eigenvalues of <math>\ \vec a \cdot \vec \sigma\ </math> are <math>\ \pm |\vec{a}|.</math> This follows immediately from tracelessness and explicitly computing the determinant.

More abstractly, without computing the determinant, which requires explicit properties of the Pauli matrices, this follows from <math>\ (\vec a \cdot \vec \sigma)^2 - |\vec a|^2 = 0\ ,</math> since this can be factorised into <math>\ (\vec a \cdot \vec \sigma - |\vec a|)(\vec a \cdot \vec \sigma + |\vec a|)= 0.</math> A standard result in linear algebra (a linear map that satisfies a polynomial equation written in distinct linear factors is diagonal) means this implies <math>\ \vec a \cdot \vec \sigma\ </math> is diagonal with possible eigenvalues <math>\ \pm |\vec a|.</math> The tracelessness of <math>\ \vec a \cdot \vec \sigma\ </math> means it has exactly one of each eigenvalue.

Its normalized eigenvectors are 
<math display="block">
  \psi_+ = \frac{1}{\sqrt{2 \left|\vec{a} \right|\ (a_3+\left|\vec{a}\right|)\ }\ } \begin{bmatrix} a_3 + \left|\vec{a}\right| \\ a_1 + ia_2 \end{bmatrix}; \qquad
  \psi_- = \frac{1}{\sqrt{2|\vec{a}|(a_3+|\vec{a}|)}} \begin{bmatrix} ia_2 - a_1 \\  a_3 + |\vec{a}| \end{bmatrix} ~ .
</math>
These expressions become singular for <math>a_3\to 
 -\left|\vec{a} \right|</math>. They can be rescued by letting <math>\vec{a}=\left|\vec{a} \right|(\epsilon,0,-(1-\epsilon^2/2))</math> and taking the limit <math>\epsilon\to0</math>, which yields the correct eigenvectors (0,1) and (1,0) of <math>\sigma_z</math>.

Alternatively, one may use spherical coordinates <math>\vec{a}=a(\sin\vartheta\cos\varphi, \sin\vartheta\sin\varphi, \cos\vartheta)</math> to obtain the eigenvectors <math>\psi_+=(\cos(\vartheta/2), \sin(\vartheta/2)\exp(i\varphi))</math> and <math>\psi_-=(-\sin(\vartheta/2)\exp(-i\varphi), \cos(\vartheta/2))</math>.