## Derivative w/respect to Anti-symmetric matrix

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.

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This question appears more appropriate for math.stackexchange.com. But you should be able to work it all out yourself comparing the definition of a derivative with respect to a "structured" matrix to the derivative with respect to an unstructured matrix. – Deane Yang Aug 29 2011 at 14:26