# Is this a typo in Peskin's QFT?

In ''An intro to QFT (2018)'' chapter 3, Peskin does the following:

Let me introduce some notation first, let $$v^s_k=\begin{pmatrix}\;\;\,\sqrt{k\cdot\sigma}\,\xi^{-s}\\-\sqrt{k\cdot\bar{\sigma}}\,\xi^{-s}\end{pmatrix}$$ be a bispinor for the negative-energy solution to the Dirac eq. with momentum $$k$$ and spin state $$\xi^{-s}$$.

From previous chapters we know $$\,\xi^{-s}\equiv-i\sigma_2(\xi^s)^*$$ and $$\,\sigma_2^*=-\sigma_2\,$$, then, $$\,(\xi^{-s})^*=-i\sigma_2\,\xi^s$$. We also know $$\,(\sqrt{k\cdot\sigma^*}\sigma_2=\sigma_2\sqrt{k\cdot\bar{\sigma}}\,)\,$$ and $$\,(\sqrt{k\cdot\bar{\sigma}^*}\sigma_2=\sigma_2\sqrt{k\cdot\sigma}\,)$$ so we can compute $$(v^s_k)^*$$ as

$$(v^s_k)^*=\begin{pmatrix}-i\sqrt{k\cdot\sigma^*}\sigma_2\,\xi^s\\\;\;i\sqrt{k\cdot\bar{\sigma}^*}\sigma_2\,\xi^s\end{pmatrix}=\begin{pmatrix}-i\sigma_2\sqrt{k\cdot\bar{\sigma}}\,\xi^s\\\;\;i\sigma_2\sqrt{k\cdot\sigma}\,\xi^s\end{pmatrix}=\begin{pmatrix}0 & -i\sigma_2 \\ i\sigma_2 & 0\end{pmatrix} \begin{pmatrix}\;\;\,\sqrt{k\cdot\sigma}\,\xi^s\\-\sqrt{k\cdot\bar{\sigma}}\,\xi^s\end{pmatrix}=-i\gamma^2 u^s_k$$.

Here, $$u^s_k$$ is the bispinor of the positive-energy solution with momentum $$k$$ and spin state $$\xi^s$$.

We see that $$\:\boxed{\,(v^s_k)^*=-i\gamma^2 u^s_k\,}\;$$ but my question comes now as Peskin proceeds saying that the following expressions follow immediately from it:

$$u^s_k=-i\gamma^2 (v^s_k)^*\;$$ and $$\;v^s_k=-i\gamma^2 (u^s_k)^*$$.

How is that possible? They don't even hold for $$\,u^s_k\neq0$$. Did I missed something?

Schwartz uses negative signature, so $$(-i\gamma^2)^2=1$$.