= Solution
Use the common <eigenvectors> of $A$ and $A^{\mathsf T}$. For each $k$,
$$
S\mathbf v_k=\frac12(\lambda_k+\lambda_k^{-1})\mathbf v_k
=\cos\frac{2\pi k}{3}\,\mathbf v_k.
$$
Thus, without constructing $S$ explicitly, its <eigenvalues> are
$$
\boxed{1,\ -\frac12,\ -\frac12.}
$$
The <eigenvalue> $1$ has direction $(1,1,1)^{\mathsf T}$. The repeated <eigenvalue> $-1/2$ has a real <eigenspace> consisting of <vectors> whose coordinates sum to zero. This is the general rule for the <symmetric part of an orthogonal matrix>. Adding unrelated lists of <eigenvalues> would not suffice; here the shared <eigenvectors> justify the pairing.
Back to article page
