Calculate the Hessian of a Vector Function

This is something many multivariate calculus courses often miss. Let's recall where the derivative lives. If a function $F\colon V\to W$ where $V$ and $W$ are finite dimensional vector spaces, then $DF\colon V\to L(V,W)$. Using this, let's figure out where the second derivative of a vector function lives. If we have $f\colon \mathbb{R}^{n}\to\mathbb{R}^{n}$ then $DF\colon \mathbb{R}^{n}\to L(\mathbb{R}^{n},\mathbb{R}^{n})$. Thus, $D^{2}F\colon \mathbb{R}^{n} \to L(\mathbb{R}^{n},L(\mathbb{R}^{n},\mathbb{R}^{n}))$. Thus, $D^{2}F(x)\in L(\mathbb{R}^{n},L(\mathbb{R}^{n},\mathbb{R}^{n}))$.

Thus, the hessian of a vector valued function can be thought of a vector of matrices. For instance, one can verify that provided $F\in C^{3}$, $D^{2}F(x)\cdot e_{i} = H_{f_{i}}(x)$.

If you want more education on this matter, I recommend Cartan's Differential Calculus. The book was recently reprinted.

The hessian of a vector valued function is a 3-tensor, which is simply a trilinear form. So $H \vec{F}=\dfrac{\partial F_i}{\partial x_k\partial x_j}$ and we have that $H \vec{F}(\vec{v},\vec{w},\vec{u})=\sum v_i \vec{u}H F_i \vec{w}$