I was trying to find a proof of the bordered hessian test for optimization problems with constraints but the only thing I found was:
z' H z <= 0 for all z satisfying Σi gi zi = 0 where H is the bordered Hessian and gi are the partial derivative of the constraint g=0. This requires that the border-preserving principle minors determinants alternate in sign. I.e., the 3 x 3 determinant (including the border) is positive, the 4 x 4 determinant is negative, and so on.
Where can I find a proof of this test or how do I proceed to prove this test?