Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof resit / 5 / d / Solution

Let $Av=\lambda v$ for a real eigenvalue $\lambda$ and a nonzero real eigenvector $v$. Part (b) gives $\lambda \ne 0$ and $A^{-1}v={\lambda }^{-1}v$. The expression for $B$ in part (c) therefore givesConsequently $\lambda +1+{\lambda }^{-1}$ is a real eigenvalue of $B$, with the same eigenvector $v$. For a real matrix and real eigenvalue, a nonzero real vector can be chosen in the real kernel of $A-\lambda I$; no complex-vector assumption is needed.