Suppose $x\in\mathbb{R}^n$, $s(x)\in\mathbb{R}^n$ and $r(x)\in\mathbb{R}^n$, being $s$ and $r$ smooth mappings. I know the following property holds: $$ \dfrac{\partial}{\partial x} s(x)^\intercal A \,r(x) = \left[\dfrac{\partial s(x)}{\partial x}\right]^\intercal A \, r(x) + \left[\dfrac{\partial r(x)}{\partial x}\right]^\intercal A^\intercal \, s(x) \quad\quad\text{with $A\in\mathbb{R}^{n\times n}$.}$$
I'm having troubles in finding a generic expression for the Hessian $\mathscr{H}$, i.e. $$\mathscr{H} = \dfrac{\partial^2}{\partial x^2} s(x)^\intercal A \,r(x)$$
Any help will be much appreciated! Thanks in advance!
Edit: I'm still not sure if the correct definition is $$\mathscr{H} = \dfrac{\partial^2}{\partial x^2} s(x)^\intercal A \,r(x) \quad \text{or} \quad \mathscr{H} = \dfrac{\partial^2}{\partial x \partial x^{\intercal}} s(x)^\intercal A \,r(x)$$ as in https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf Equation (98).
Assuming that you can calculate the following matrix and (third-order) tensor derivatives of the mappings. $$ S = \frac{\partial s}{\partial x}, \quad{\mathbb S}= \frac{\partial S}{\partial x} = \frac{\partial^2s}{\partial x^2} \quad\implies\quad S_{ij}= \frac{\partial s_i}{\partial x_j}, \quad{\mathbb S}_{ijk} = \frac{\partial S_{ij}}{\partial x_k} = \frac{\partial^2s_i}{\partial x_j\partial x_k} = \frac{\partial^2s_i}{\partial x_k\partial x_j} \\ R = \frac{\partial r}{\partial x}, \quad{\mathbb R}= \frac{\partial R}{\partial x} = \frac{\partial^2r}{\partial x^2} \quad\implies\quad R_{ij}= \frac{\partial r_i}{\partial x_j}, \quad{\mathbb R}_{ijk} = \frac{\partial R_{ij}}{\partial x_k} = \frac{\partial^2r_i}{\partial x_j\partial x_k} = \frac{\partial^2r_i}{\partial x_k\partial x_j} \\ $$ Find the differential and gradient of the cost function. $$\eqalign{ \phi &= A:sr^T \\ d\phi &= A:(s\,dr^T + ds\,r^T) \\ &= A^Ts:dr + Ar:ds \\ &= A^Ts:R\,dx + Ar:S\,dx \\ &= (R^TA^Ts + S^TAr):dx \\ \frac{\partial \phi}{\partial x} &= R^TA^Ts + S^TAr \;=\; g \\ }$$ which confirms the cookbook result.
Now calculate the gradient of $g$ (aka the hessian of $\phi$). $$\eqalign{ dg &= dR^TA^Ts + S^TA\,dr + R^TA^Tds + dS^TAr \\ &= ({\mathbb R}\,dx)^TA^Ts +S^TA(R\,dx) +R^TA^T(S\,dx) +({\mathbb S}\,dx)^TAr \\ &= (s^TA\,{\mathbb R} + S^TAR + R^TA^TS + r^TA^T{\mathbb S})\,dx \\ \frac{\partial g}{\partial x} &= (s^TA\,{\mathbb R} + S^TAR + R^TA^TS + r^TA^T{\mathbb S}) \;=\; \frac{\partial^2 \phi}{\partial x^2} \;=\; {\scr H} \\ }$$