Product rule of the derivative of a matrix by a vector

Question

I am trying to express the derivative of the outer product of the $(n\times m)$-matrix $\mathbf{A}=\mathbf{A}(\mathbf{x})$ with respect to the $p$-vector $\mathbf{x}$. This is, I want to rewrite $\frac{\partial \mathbf{A}\mathbf{A}^T}{\partial \mathbf{x}}$ using a product rule. My intuition tells me that I must have something like $$ \frac{\partial \mathbf{A}\mathbf{A}^T}{\partial \mathbf{x}}=\mathbf{A}\otimes\frac{\partial \mathbf{A}}{\partial \mathbf{x}}+\frac{\partial \mathbf{A}}{\partial \mathbf{x}}\otimes\mathbf{A}. $$ Any help or confirmation on this? Thanks!

Answer 1

Note $f(A)=AA^T$. $f$ is the function composition of $A \mapsto (A,A^T)$ which is linear and of $(A,B) \mapsto AB$ which is bilinear.

Hence $$f^\prime(A).h=h.A^T+A.h^T$$

Finally if $A$ depends on a variable $x$ and applying the chain rule, you have $$\frac{d(AA^T)}{dx}=\frac{dA}{dx}A^T +A(\frac{dA}{dx})^T$$