 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-331/4/a/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

The relevant norm is the operator norm induced by the complex Euclidean norm, also called the matrix 2-norm:

∥ B ∥_{2} = q \neq = 0 max \frac{∥ Bq ∥ _{2}}{∥ q ∥ _{2}} = ∥ q ∥_{2} = 1 max ∥ Bq ∥_{2} .

The maximum exists because the unit sphere in finite dimensions is compact. It obeys

∥ Bq ∥_{2} \leq ∥ B ∥_{2} ∥ q ∥_{2}

and is the norm relevant to the specified perturbation energy. Other matrix norms do not in general give this gain.

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