= Solution
Induction on $p$ gives $S^p\mathbf u_i=\lambda_i^p\mathbf u_i$, including $p=0$. The absolutely convergent <matrix exponential> series may therefore be applied term by term to each <eigenvector>:
$$
C\mathbf u_i=\sum_{p=0}^\infty\frac{S^p\mathbf u_i}{p!}
=\left(\sum_{p=0}^\infty\frac{\lambda_i^p}{p!}\right)\mathbf u_i
=e^{\lambda_i}\mathbf u_i.
$$
Thus
$$
\boxed{\text{eigenvectors }\mathbf u_i,\qquad \text{eigenvalues }e^{\lambda_i}}.
$$
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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