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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?

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    This seems to be how to define $\mathrm{d}$ on a matrix: $\mathrm{d}(a_{ij})=(\mathrm{d}a_{ij})$. That seems to give what you want, does it not?2013-10-20

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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.

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    Any operation which merely rearranges the elements of $X$, (e.g. vec, transpose, reshape, vecpose, block-vecpose) satisfies the differential property. It rearranges the elements of $dX$ in the same way.2015-03-22