Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 30L / a

Table of contents
- Solution a

Solution

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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