Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 6 / f

For a real eigenvalue $\lambda$, choose a nonzero real eigenvector $v$ with $Av=\lambda v$. By (e),Since $v^{\top }v=\Vert v{\Vert }^2>0$, $\lambda =0$. If eigenvectors were initially allowed to be complex, the real matrix $A-\lambda I$ has a nonzero real null space whenever it has a nonzero complex null space, so this choice of $v$ remains justified.