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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?

2 Answers 2