[Math] 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 $1 /\sqrt{2}$.