I read in the matrix cookbook, section 2.8, that rules for differentiation of a scalar function of a matrix differ depending on whether that matrix has a structure (e.g. symmetric, diagonal, etc).
Does this extend to matrix functions? For example, for an unstructured matrix $X$,
$$ \frac{d(AXB)}{dX}=B^T\otimes A $$
where $\otimes$ is the kronecker product. But if $X$ is anti-symmetric, does this then become
$$ \frac{d(AXB)}{dX}=B^T\otimes A-(B^T\otimes A)^T $$
This is my guess since in the case of a scalar function $f(X)$,
$$ \frac{df(X)}{dX}=\left[\frac{df(X)}{dX}\right]-\left[\frac{df(X)}{dX}\right]^T. $$
Please say if my question is unclear. Thanks.
