= Solution
If $A$ is a real <symmetric matrix>, the finite-dimensional spectral theorem lets us choose the normalized <left eigenvector> and <right eigenvector> to be the same vector. Thus
$$
|u^Tv|=1,\qquad \boxed{s(\lambda)=1}.
$$
More generally, the <unitary diagonalization of a normal matrix> shows that a real <normal matrix> also has coincident normalized left and right eigenspaces for each real eigenvalue. Since the question assumes $\lambda\in\mathbb R$, one can again choose $u=v$, and every such simple eigenvalue has
$$
\boxed{s(\lambda)=1}.
$$
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