 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ia/paper-1/7b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

From

H v = μv

v^{†} H v = μ ∥ v ∥^{2}

is real, so

μ \in R

. For distinct eigenvalues,

μ_{i} v_{i}^{†} v_{j} = v_{i}^{†} H v_{j} = μ_{j} v_{i}^{†} v_{j}

, hence orthogonality. Normalising the eigenvectors and taking them as the rows of

U

gives

H = U^{†} Λ U

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