What does it mean for the hessian to have a value? The hessian matrix is the matrix formed by taking the second derivatives of some vector $X$.
$$\nabla^2 X = H$$
In determining whether a function at critical point is a minimum or maximum, we test to see whether the hessian is positive or negative.
Am I correct that to check whether the hessian is positive, we simply check whether all entries of the matrix are $> 0$?
 A: (Note: The Hessian (as an operator) is not applied to "some vector $X$".  The Hessian is applied to a scalar function.)
I assume you're talking about the second derivative test.  In that case what you're looking for is whether the Hessian matrix is positive definite, negative definite, indefinite but nondegenerate, or none of the above.
Definition: A real-valued symmetric, square matrix is positive (negative) definite if all of its eigenvalues are positive (negative) real numbers.
The second derivative test states:


*

*If your Hessian is positive definite at a critical point then that
point is a local minimum of your function.

*If your Hessian is negative definite at a critical point then that
point is a local maximum of your function.

*If your Hessian has positive and negative eigenvalues (but no
zeros) at a critical point, then that point is a saddle point.

*If your matrix has a zero eigenvalue at a critical point then the test is inconclusive.



As some schools require multivariate calculus before linear algebra, if you don't understand the concept of an eigenvalue let me know and I'll give you an alternate criterion for determining positive/ negative definiteness.
A: Just look at the eigenvalues. If they are strictly positive, then you say that the Hessian (or matrix in general) is positive definite.
