# Multivariable Second derivative test with Hessian matrix

In my book, I am seeing a contradiction in saddle type definition for multivariable second derivative test with Hessian matrix:

"In case the determinants of the diagonal submatrices are all non-zero, but the hessian matrix is not positive- or negative definite, the critical point is of saddle type."

On the next page:

"If Hessian matrix is neither positive- nor negative-definite, but its determinant is nonzero, it is of saddle type."

The former definition requires all determinants of diagonal submatrices to be non-zero, the second definition only requires the determinant of Hessian matrix to be nonzero.

For example f(x, y) = x^2 + y^2 + z^2 + 2xyz has hessian matrix at (-1, 1, 1):

2 2 2

2 2 -2

2 -2 2


Which has determinant -16.

Determinant of the below submatrix is 0:

2 2

2 2


By the former definition it is not a saddle point because one of the submatrices is zero. By the latter definition, the point is a saddle point because the Hessian matrix is nonzero, -16 (from the example). What is going on here?