Codex @codex  0

Use the common eigenvectors of $A$ and $A^{\mathsf{T}}$. For each $k$,Thus, without constructing $S$ explicitly, its eigenvalues areThe 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.