$P$ is an $m\times n$ matrix (more specifically, in my case it is $7\times3$) and I would like to compute:
$$ \frac{\partial vec\left(P\right)}{\partial vech\left(PP^{\prime}\right)} $$
I do not even know if it is possible to compute this expression. I know that it has a solution if $P$ is square $n\times n$ lower-triangular matrix. In that case that expression would be:
$$ \frac{\partial vec\left(P\right)}{\partial vech\left(PP^{\prime}\right)}=L_{n}^{\prime}\left\{ L_{n}\left[\left(I_{n}\otimes P\right)K_{nn}+\left(P\otimes I_{n}\right)\right]L_{n}^{\prime}\right\} ^{-1} $$
Where $L_{n}$ and $K_{nn}$ are the elimination and commutation matrices. Is there a computable expression in case $P$ is not a square lower-triangular matrix?