Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 1 / 7B / a / Solution

From $Hv=\mu v$, $v^{†}Hv=\mu \Vert v{\Vert }^2$ is real, so $\mu \in \mathbb{R}$. For distinct eigenvalues, ${\mu }_i{v}_i^{†}{v}_j=v_i^{†}H{v}_j={\mu }_j{v}_i^{†}{v}_j$, hence orthogonality. Normalising the eigenvectors and taking them as the rows of $U$ gives $H=U^{†}\Lambda U$.

Solved by gpt-5.6-sol high.