I wanted to add a comment to the following:
but I don't have enough reputation for that.
Both answers are too short for me to understand how to compute the Jacobian of a matrix and/or a Hessian of a vector.
The only thing I understood is that the final result should be a vector of matrices, but it is not clear if it's a column vector or a row vector, in which order it should be computed, and what dimension do the matrices have.
Can somebody show a complete example with a very small matrix/vector? e.g.: $$ \frac{\partial{\mathbf{A}(\boldsymbol{\alpha})}}{\partial \boldsymbol{\alpha}} = \begin{bmatrix} \begin{bmatrix} \frac{\partial a_{11}}{\partial{\alpha_{11}}} & \dots & ? \\ \dots &\ddots & ? \end{bmatrix} \\ \begin{bmatrix} \frac{\partial a_{??}}{\partial{\alpha_{??}}} & \dots & ? \\ \dots &\ddots & ? \end{bmatrix}\\ \dots \end{bmatrix} \quad\boldsymbol{\alpha} \in \mathbb{R}^3,\;\mathbf{A}(\boldsymbol{\alpha}) :\mathbb{R}^3\mapsto\mathbb{R}^{2\times3} $$ and same for the Hessian of a small vector valued function, e.g. $\mathbf{f}(\boldsymbol{\alpha}) : \mathbb{R}^3 \mapsto \mathbb{R}^2$:
