 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/18/b/ii/solution

Induction on

p

gives

S^{p} u_{i} = λ_{i}^{p} u_{i}

, including

p = 0

. The absolutely convergent matrix exponential series may therefore be applied term by term to each eigenvector:

C u_{i} = \sum_{p = 0}^{\infty} \frac{S ^{p} u _{i}}{p !} = (\sum_{p = 0}^{\infty} \frac{λ _{i}^{p}}{p !}) u_{i} = e^{λ_{i}} u_{i} .

Thus

eigenvectors u_{i}, eigenvalues e^{λ_{i}} .

For a repeated eigenvalue, every nonzero vector in its eigenspace remains an eigenvector. Because the real exponential function is injective, distinct real eigenvalues of

S

do not merge under this transformation.

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