Derivative of vectorized kronecker product

Question

I'm struggling with the following derivative. Let $\pmb{X}$ be a symmetrical $n \times n$ matrix, $\pmb{x} = \mathrm{vec}(\pmb{X})$ and let function $\pmb{f}$ take the following form: $$ \pmb{f} = \mathrm{vec}\left( \pmb{X} \otimes \pmb{I}_n \right) $$ This is part of a vectorized Jacobian for which I am trying to derive a Hessian. I want to find a solution for: $$ \frac{\partial \pmb{f}}{\partial \pmb{x}} $$ I recognized that there is some $n^4 \times n^2$ matrix $\pmb{Y}$, likely containing only zeroes and ones, such that: $$ \mathrm{vec}\left( \pmb{X} \otimes \pmb{I}_n \right) = \pmb{Y} \pmb{x}, $$ as $\pmb{f}$ just contains only the elements of $\pmb{x}$ multiple times including a lot of zeroes. This would automatically imply: $$ \frac{\partial \pmb{f}}{\partial \pmb{x}} = \pmb{Y} $$ However, I can't think of a logical structure for $\pmb{Y}$.

Answer 1

Let {$e_k$} be the standard vector basis, or if you prefer, the columns of $I_n$.

Then $$\eqalign{ M &= \pmatrix{I_n\otimes e_1\cr I_n\otimes e_2\cr \vdots \cr I_n\otimes e_n} \cr \frac{\partial f}{\partial x} = Y &= \Big(I_n\otimes M\Big) \cr }$$ A related problem is $$\eqalign{ g &= {\rm vec}(I_n\otimes X) \cr \frac{\partial g}{\partial x} &= \Big(M\otimes I_n\Big) \cr }$$

Update

Using a Commutation matrix provides the compact expression $$M = \big(K^{(n,n)}\otimes I_n \big)\:\big(I_n\otimes{\rm vec}(I_n)\big)$$