Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 5 / b

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- Solution b

Solution

Use a singular value decomposition

X = U D V^{T}

. Then

(X^{T} X + λ I)^{- 1} X^{T} = V (D^{T} D + λ I)^{- 1} D^{T} U^{T} .

Each nonzero singular value

d

contributes

d / (d^{2} + λ) \to d^{- 1}

, while each zero one contributes zero. Thus

lim_{λ ↓ 0} β_{λ} = V D^{+} U^{T} Y = (X^{T} X)^{+} X^{T} Y,

using the Moore-Penrose inverse.

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