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.