 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-4/30l/a/solution

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

The candidate

Π

is tacitly required to belong to

S

, as in the usual variational characterization of convex projection; this feasibility holds for the candidate constructed in part (b). Use the Frobenius inner product

⟨ A, B ⟩_{F} = Tr (A^{T} B) .

For any

Z \in S

M - Z = (M - Π) - (Z - Π),

and hence

∥ M - Z ∥_{F}^{2} = ∥ M - Π ∥_{F}^{2} + ∥ Z - Π ∥_{F}^{2} - 2 ⟨ M - Π, Z - Π ⟩_{F} \geq ∥ M - Π ∥_{F}^{2} + ∥ Z - Π ∥_{F}^{2} \geq ∥ M - Π ∥_{F}^{2} .

Thus

Π

minimizes the squared Frobenius distance over

S

. Since that objective is strictly convex,

Π = π (M) .

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