 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2023/4/a/solution

∥ exp A - I - A ∥ \leq \sum_{j = 2}^{\infty} \frac{∥ A ∥ ^{j}}{j !} = e^{∥ A ∥} - 1 - ∥ A ∥ = O (∥ A ∥^{2}) .

Therefore

∥ exp A - exp 0 - A ∥/∥ A ∥ \to 0

. This is precisely Fréchet differentiability at zero, with

D exp (0) [H] = H .

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