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I'm trying to calcaute the eigenstates for the $\sigma_x$ gate, and I can follow the process up to finding eigenvalues $\pm 1$, but I don't understand where the $\frac{1}{\sqrt{2}}$ coefficient comes from for the answer:

$$\begin{bmatrix} -\lambda & 1\\ 1 & -\lambda \end{bmatrix}v = 0 \implies v = \frac{1}{\sqrt{2}} \begin{bmatrix} 1\\ 1 \end{bmatrix} $$

For the solution $\lambda = 1$, why does that $\frac{1}{\sqrt{2}}$ show up?

1Forgot about the normalization part. Cheers! – Markian Hromiak – 2021-01-11T06:23:35.690