# Spinor rotation around the $z$-axis

In Schwartz's QFT textbook (eq. 10.118), he gives the equation for the Lorentz transformation matrix of a rotation around the $z$-axis for as spinor as $$\Lambda_s(\theta_z)=\left( \begin{array}{cccc} \exp({i\over 2}\theta_z)& & & \\ &\exp({-i\over 2}\theta_z) & & \\ & & \exp({i\over 2}\theta_z)& \\ & & & \exp({-i\over 2}\theta_z)\\ \end{array} \right) \tag{10.118}$$ He then states on pg 176 that $\Lambda_s(\pi)=i$ for each spinor. I don't understand this as from the above equation shouldn't it be: $$\Lambda_s(\pi)=\left( \begin{array}{cccc} i& & & \\ &-i& & \\ & & i& \\ & & & -i\\ \end{array} \right)$$

$\Lambda_s(\pi)$ is certainly not $i$ in general. However, on that page Schwartz applies this to the z-spin up particles. In which case it acts as multiplying by $i$.