What is the difference between:
1) $\frac{\partial (\textbf{x}^{T}A)}{\partial \textbf{x}}$
and
2) $\frac{\partial (A\textbf{x})}{\partial \textbf{x}}$
where A is a nxn matrix and x is a n sized column vector.
Using the definition of a Jacobian on wikipedia (http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant)
My answers are:
1) $A^{T}$
2) $A$
However this is not correct because:
$$\textbf{x}^{T}A = (A^{T}\textbf{x})^{T}$$
performing $\frac{\partial}{\partial \textbf{x}}$ on each side results
$$A^T=A$$
which is not true. Does $\frac{\partial}{\partial \textbf{x}}$ become $\frac{\partial}{\partial \textbf{x}^{T}}$ when moved inside the transpose on the right side?