# Worked Problems in Physics

## Nielsen and Chuang Exercise 2.60

Posted in Nielsen/Chuang by rpmuller on March 23, 2010

Exercise 2.60 Show that ${(\vec v\cdot\vec\sigma)}$ has eigenvalues ${\pm 1}$, and that the projectors on to the corresponding eigenspace are given by ${P_{\pm}=(I\pm\vec v\cdot\vec\sigma)/2}$.

Starting from the spectral decomposition

$\displaystyle U^\dagger(\vec v\cdot\vec\sigma)U=\Lambda$

and squaring both sides yields

$\displaystyle \Lambda^2=U^\dagger(\vec v\cdot\vec\sigma)UU^\dagger(\vec v\cdot\vec\sigma)U =U^\dagger(\vec v\cdot\vec\sigma)(\vec v\cdot\vec\sigma)U=U^\dagger IU=I.$

(The second-to-last step uses the result from Exercise 2.35 that ${(\vec v\cdot\vec\sigma)(\vec v\cdot\vec\sigma)=I}$.) Since the eigenvalues of the Hermitian matrix ${(\vec v\cdot\vec\sigma)}$ are real, ${\lambda_i=\pm 1}$.

We can easily verify that ${P_{\pm}=(I\pm\vec v\cdot\vec\sigma)/2}$ meets the requirements for the project that ${\sum_m P_m=I}$, that ${P_mP_n=\delta_{mn}P_m}$, and that ${M=\sum_m mP_m}$. Thus, ${P_{\pm}=(I\pm\vec v\cdot\vec\sigma)/2}$ is a set of projectors for ${(\vec v\cdot\vec\sigma)}$.