What conditions have to hold for the Hessian of a function to be symmetric positive definite?
A matrix is called symmetric positive definite if all its eigenvalues are greater than or equal to zero. The following necessary and sufficient condition holds for a function to have a Hessian that is symmetric positive definite:

1.  The function is twice differentiable at every point in its domain.
2.  The second derivative of the function, denoted by ∂^2 f/∂x^2, is positive semidefinite at each point in its domain.
3.  At any critical point (i.e., where the first derivative is 0), the Hessian must be nonsingular.
4.  For any two functions f and g whose domains overlap, their mixed second derivatives ∂^2f/∂x^2 + ∂^2g/�partialy^2 are always positive semidefinite.
5.  If a function is convex, then its Hessian will be positive definite.
6.  Any multiple of a positive-definite matrix is also positive definite.
7.  The Hadamard product of two matrices is also positive definite.
8.  The Frobenius norm of a matrix measures its largest singular value, and it satisfies the inequality |H| \le 1 for any Hermitian matrix H.
9.  Every positive definite matrix can be written as the sum of three simpler positive definite matrices, known as Schur complements.
10. A symmetric positive definite matrix has real entries.