TITLE: Differential of transposed matrices
QUESTION [6 upvotes]: I'm puzzling about how to deal with the differential of a transposed matrix.
I was wondering if there is some rule such that $d(X^{T}) = (dX)^{T}$.
In general I work with derivation on the trace of a matrix and I get sometimes the following situation:
$$ tr(d(X^{T})AX + Bd(X^{T})CX + DdX) $$
where X can be a rectangular matrix.
I'm quite sure that such expression can be rearranged as follows:
$$ tr((AX + CXB)d(X^{T})) + tr(DdX) $$
Clearly I would like to obtain something like $tr(J(X)dX)$ for derivative, but I'm not able to go on.
Some suggestion?

REPLY [4 votes]: The relation $d(X^T) = d(X)^T$ is true.
You can easily verify this by letting $X$ be a square matrix and then by comparing the left and right hand side of above the relation.