Normalizing Eigenvectors from Pauli Matrices

Question

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)$

Answer 1

The norm of the eigenvector is $$||\begin{bmatrix} -i \\ 1\end{bmatrix}|| =\sqrt{|-i|^2+|1|^2}=\sqrt{2},$$ therefore the normalization constant is $1 /\sqrt{2}$.

Answer 2

Most likely your mistake is treating the norm as $$\sqrt{(-i)^2+1^2} = 0$$

It should be $$\sqrt{|-i|^2+|1|^2} = \sqrt{2}$$