Differentiate matrix product with respect to vectorized form

Question

I am very new to matrix calculus, but I can't seem to figure this out. Suppose $\pmb{X}$ is a non-square $m \times n$ matrix and $\pmb{Y}$ is a symmetrical positive definite $n \times n$ matrix that is not a function of $\pmb{X}$. I need to differentiate: $$ \frac{\partial }{\partial \mathrm{vec}\left( \pmb{X} \right)} \mathrm{vec}\left( \pmb{X} \pmb{Y} \pmb{X}^{\top} \right) $$ which should lead to a $n^2 \times nm$ matrix I think. My first thought was to make use of $\mathrm{vec}\left( \pmb{X} \pmb{Y} \pmb{X}^{\top} \right) = (\pmb{X} \otimes \pmb{X}) \mathrm{vec}(\pmb{Y})$, but post-multiplying $\mathrm{vec}(\pmb{Y})$ with any result of differentiating $\pmb{X} \otimes \pmb{X}$ with respect to $\mathrm{vec}\left( \pmb{X} \right)$ doesn't seem to give a reult of the right dimensions.

Answer 1

Consider the vector $$\eqalign{ w &= {\rm vec}(XYZ) \cr &= (Z^TY^T\otimes I)\,{\rm vec}(X) &= (Z^T\otimes X)\,{\rm vec}(Y) &= (I\otimes XY)\,{\rm vec}(Z) \cr &= (Z^TY^T\otimes I)\,x &= (Z^T\otimes X)\,y &= (I\otimes XY)\,z \cr }$$ and its differential $$\eqalign{ dw &= (Z^TY^T\otimes I)\,dx + (Z^T\otimes X)\,dy + (I\otimes XY)\,dz \cr }$$ Now assume $Y$ is constant and $Z=X^T$, i.e. $$dy=0,\quad dz=K\,dx$$ where $K$ is the Commutation Matrix associated with the vec-operation.

Substituting this into the differential expression yields $$\eqalign{ dw &= \Big(\big(XY^T\otimes I\big) + \big(I\otimes XY\big)K\Big)\,dx \cr \frac{\partial w}{\partial x} &= \big(XY^T\otimes I\big) + \big(I\otimes XY\big)K \cr }$$