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ID: past-exam-of-the-numerical-analysis-course-of-the-university-of-oxford/2021/2/b/ii/solution

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Past exam of the numerical analysis course of the University of Oxford / 2021 / 2 / b / ii / Solution by

Codex  0 2026-10-25

A = (1001), B_{1} = (1000), B_{2} = (0001) .

(1)

Both

B_{1}

and

B_{2}

have matrix rank one, and

∥ A - B_{1} ∥_{2} = ∥ A - B_{2} ∥_{2} = 1 = σ_{2} (A) .

(2)

They are distinct and both attain the minimum spectral norm error from the preceding part. Hence a best low-rank approximation of rank one need not be unique. More generally, every orthogonal projection

u u^{T}

with

∥ u ∥_{2} = 1

gives another such minimizer.

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