Is there an example of an $n \times n$ matrix that:
- is real-valued and symmetric with entries between $-1$ and $+1$
- has diagonal elements equal to $1$
- has a non-negative determinant
- has non-negative determinant for each leading principal minor
but is not a correlation matrix, i.e., is not positive semidefinite?
I am well-aware of Sylvester's criterion for positive definiteness, which requires all principal minors to be non-negative to ensure positive semidefiniteness. However, for structured matrices of the specified format (correlation matrix type), I never came across an actual example illustrating the subtle difference.