I am looking for the minimum additional restrictions on a real symmetric $(n\times n)$ matrix $A$, such that its eigenvalues are invariant under a sign change of all off-diagonal elements.
I don't know if that's a clever way to phrase it, but more formally: $$A\text{ is real symmetric and [...?]}\Longleftrightarrow\det{(A-I\lambda)}=\det{(A\circ(2I-J)-I\lambda)}$$ where $\circ$ is the element-wise (Hadamard) product and $J$ is the matrix of ones.
Most simple cases:
- For $n=2$, the statement is true for any real symmetric matrix (any complex matrix indeed).
- For $n=3$, simplification of the equation yields $A_{12}A_{13}A_{23}=0$, which means the eigenvalues are invariant, if at least one of the off-diagonal elements are zero.
- For $n=4$, variable $\lambda$ does not cancel out during simplifciation(?), so I'm somewhat stuck with the straightforward approach.