For this example of a Pauli matrix, \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} I found that one of its eigenvectors (for $\lambda = 1$) is \begin{bmatrix} -i \\ 1 \end{bmatrix} but when I try to normalize it, I run into the issue of dividing by zero. Can anyone guess what I'm doing wrong here?
My problem setup: $ \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right) % \left( \begin{array}{cc} x \\ y \end{array} \right) % = (+1) \left( \begin{array}{cc} x \\ y \end{array} \right)$
$\endgroup$ 42 Answers
$\begingroup$The norm of the eigenvector is $$||\begin{bmatrix} -i \\ 1\end{bmatrix}|| =\sqrt{|-i|^2+|1|^2}=\sqrt{2},$$ therefore the normalization constant is $1 /\sqrt{2}$.
$\endgroup$ 7 $\begingroup$Most likely your mistake is treating the norm as $$\sqrt{(-i)^2+1^2} = 0$$
It should be $$\sqrt{|-i|^2+|1|^2} = \sqrt{2}$$
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