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.
Yes, one simple example is $$ A=\pmatrix{1&1&1&1\\ 1&1&1&1\\ 1&1&1&-1\\ 1&1&-1&1}. $$ Obviously, the first entry of $A$ is positive, while the second, third and fourth leading principal minors of $A$ are all zero because the corresponding submatrices have at least two identical rows. However, when $x=(1,1,-2,-2)^\top$, we have $x^\top Ax=-12<0$. The trailing principal $3\times3$ minor of $A$ is negative ($-4$) in this case. If this was a covariance matrix, variables 3 and 4 would be negatively correlated, but each of them is perfectly correlated with variables 1 and 2, which is absurd.