 Solution

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

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

L = V diag (λ_{j}) V^{†}

with orthonormal eigenvectors. Thus

e^{L t} = V diag (e^{λ_{j} t}) V^{†}, ∥ e^{L t} q_{0} ∥_{2}^{2} = \sum_{j} e^{2 Re λ_{j} t} ∣ ϕ_{j} ∣^{2}, ∥ q_{0} ∥_{2}^{2} = \sum_{j} ∣ ϕ_{j} ∣^{2} .

For

t \geq 0

, the largest weight belongs to an eigenvalue with maximal real part. The weighted average is bounded by that weight and attains it on the corresponding eigenspace. Therefore the optimal energy amplification of a linear system is

G (t) = e^{2 Re λ_{1} t} (t \geq 0) .

If the largest real part is repeated, any nonzero combination within that maximal eigenspace is optimal. For negative times the ordering reverses; the displayed formula concerns forward evolution.

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