Let there be a mapping, $x \mapsto y(x)$, an SPD matrix $A$, and some constant $m \in \mathbb{R}^n$ such that the following holds:
$(y-m)^T A(y-m)/2 = x^Tx/2$.
I need to find the Jacobian of the mapping, $J = \partial y/\partial x$. Using the Matrix Cookbook, after differentiating both sides by $x$, I find
$$A(y-m)\frac{\partial y}{\partial x} = x.$$
The first question is whether I applied the chain rule correctly there. It is not clear from the sources I used. The second question is how to find $J$ given the above. From the above, the only option appears to be a scalar, which gives me pause.
Thanks for your consideration.
Using the Cholesky decomposition of the matrix one can write $$\eqalign{ A &= C^TC \\ x &= C(y-m) \\ }$$ Therefore $$\eqalign{ dx &= C\,dy \\ dy &= C^{-1}\,dx \\ \frac{\partial y}{\partial x} &= C^{-1} \\ }$$