I am looking for a characterization of functions $\Phi: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ such that $\Phi(\mathbf{x}) = \nabla^2 f(\mathbf{x})$ for a function $f$ which is twice continuously differentiable. Is there any composition rules for $\Phi$?
For example, $\Phi(\mathbf{x}) = \mathbf{A}$, where $\mathbf{A}$ is any constant symmetric matrix, is a trivial example for $\Phi$. However, I look for non-trivial examples which depend on $\mathbf{x}$. In other words, I want to find rules for $\Phi$ which can generate valid Hessian matrices.
Is this even possible? Can someone provide my correct references and pointers?
