Eigenvalues of a positive principal minors symmetric matrix

Question

I am trying to understand the proof of the Sylvesters Criterion. More concretely:

Suppose that the real symmetric matrix $A$ has only positive principal minors.

The statement that I do not understand says: "It follows that if $A$ is not positive definite, it must possess at least two negative eigenvalues.". Why can't $A$ just have one negative eigenvalue? I would be very grateful if somebody could explain this or give me a good source to read through.

Answer 1

We know that $\det(A) > 0$. Recall that $\det(A) = \prod_i \lambda_i$ where $\lambda_i$ are the eigenvalues of $A$. Then we cannot just have one negative eigenvalue if $A$ has a positive determinant.