Matrix derivative of $\frac{d(xA)}{dx}$

Question

My problem was to find the matrix derivative of:

$$ F =\begin{bmatrix}0&0&x_1\\0&0&x_2\end{bmatrix} $$

with respect to $x = \begin{bmatrix}x_1\\x_2\end{bmatrix}$. My first thought is to decompose F into:

$$ F = \begin{bmatrix}x_1\\x_2\end{bmatrix} \begin{bmatrix}0&0&1\end{bmatrix} $$

but then it is still unclear how to proceed, since I only have derivative of $Ax$, but not $xA$ with respect to $x$. Thanks for your help.

Answer 1

Write $F(x) =\begin{bmatrix}0&0&x_1\\0&0&x_2\end{bmatrix}$. Note that $F(x+h) -F(x) = F(h)$. Hence $F$ is its own derivative, that is, $DF(x)h = F(h)$.