I have the following tensor (takes a vector of length $m$ and returns a matrix $m \times m$):
$C(y) = A \operatorname{diag}(A^T y ) A^{-1}$
for some invertible matrix $A$ of size $m \times m$ ($y$ is of size $m \times 1$).
Let's say I have $C(y)$ (a way to compute it, etc.) for any $y$ I want - is there a way to solve for $A$? (i.e. identify what $A$ is.)