Is a correlation matrix with positive determinant PSD?

Question

Please note: I'm not interested in the difference between positive definiteness and semi-definiteness for this question.

A correlation matrix is a symmetric positive semi-definite matrix with 1s down the diagonal and off-diagonal terms $ -1 \leq M_{ij} \leq 1$.

Since a correlation matrix must be positive semi-definite, it must have a positive (or zero) determinant, but does a positive determinant imply positive definiteness? In other words, if I have a matrix with 1s down the diagonal, off-diagonals satisfying $ -1 \leq M_{ij} \leq 1$ and positive determinant, is that enough to prove that the matrix is positive definite (and thus an acceptable correlation matrix)?

Thank you.

Answer 1

The answer to your question, as I now understand it, is no. In particular, we can construct a matrix of you particular pattern with a positive determinant that fails to be positive definite.

In particular, consider the matrix

$$ M = \pmatrix{ 1&-1&-1&0&0&0\\ -1&1&-1&0&0&0\\ -1&-1&1&0&0&0\\ 0&0&0&1&-1&-1\\ 0&0&0&-1&1&-1\\ 0&0&0&-1&-1&1\\ } $$ which has eigenvalues $-1-1,2,2,2,2$

Answer 2

Three facts:

1. Symmetric semi positive definite matrix has all eigenvalues real and greater or equal zero.
2. Symmetric positive definite matrix has all eigenvalues real and greater than zero.
3. Determinant of any matrix is equal to product of all eigenvalues.

Hence, if determinant of symmetric semi positive definite matrix $A$ is nonzero, then $A$ is positive definite.

The answer is yes.

Answer 3

Slight variation to original poster's question, would the conjecture hold if the determinants of leading principal minors are non-negative? I.e.

Take a matrix with the following properties: a) real symmetric b) all entries between -1 and 1 c) diagonal elements are all equal to 1 d) determinants of leading principal minors are non-negative

Is there an example of such a matrix which is not positive semi-definite? I wasn't able to construct any such example yet.

This is very much related to Sylvester's criterion and already holds for positive definite matrices, whereas in the case of positive semi-definite matrices, all principal minors must have non-negative determinants. The conjecture (which seems what the original poster was after) really means that for such "nicely" structured matrices, the stronger property holds and we only have to check the leading principal minors?