Jacobian for matrix function involving kronecker product I would like to ask you a little help for the following problem.
Let $\Phi$ and $\Sigma$ be two $N \times N$ matrices s.t. the inverse of $(I_{N^2}-\Phi \otimes \Phi  )$ exists and $\Sigma$ is symmetric and positive definite.
Define $\Lambda=(I_{N^2}-\Phi \otimes \Phi  )^{-1} vec (\Sigma)$.
Note that $\Lambda$ is identity matrix $N ^2\times N^2$.
The Jacobian of $\Lambda$ with respect to $vec (\Sigma)$ should be easy to compute and given by
$$\frac{\partial \Lambda}{\partial vec (\Sigma)'}= \left[(I_{N^2}-\Phi \otimes \Phi  )^{-1} \right]^{T}.$$
What about $\dfrac{\partial \Lambda}{\partial vec (\Phi)'}$ ?
 A: For convenience, let me define some terms (with Latin letters so I don't have to type all those Greek letters)
$$\eqalign{
 s &= {\rm vec}(\Sigma) \cr
 F &= \Phi \cr
 a &= \Lambda \cr
 M &= (I\otimes I - F\otimes F) \cr
 dM &= -(dF\otimes F + F\otimes dF) \cr
}$$
Then the equation we are working with is
$$\eqalign{
 a &= M^{-1}s \cr
 Ma &= s \cr
}$$
Taking the differential
$$\eqalign{
 dM\,a + M\,da &= ds \cr
}$$
If we hold $M$ constant, then we can obtain your first result (let's not quibble over transposes)
$$\eqalign{
 M\,da &= ds \cr
 da &= M^{-1}\,ds \cr
 \frac{\partial a}{\partial s} &= M^{-1} \cr
}$$
If instead we hold $s$ constant, we obtain
$$\eqalign{
 dM\,a + M\,da &= 0 \cr
 M\,da &= -dM\,a \cr
 M\,da &= (dF\otimes F + F\otimes dF)\,a \cr
}$$
At this point we need a few tricky Kronecker-Vec relationships
$$\eqalign{
 {\rm vec}(A) &= a \cr
 (F\otimes dF)\,{\rm vec}(A) &= (FA^T\otimes I)\,{\rm vec}(dF) \cr
 (dF\otimes F)\,{\rm vec}(A) &= (I\otimes FA)\,{\rm vec}(dF^T) \cr
   &= (I\otimes FA)\,P\,\,{\rm vec}(dF) \cr
}$$
where $P$ is the appropriate Kronecker Permutation for a matrix with the dimensions of $F$.
Substituting the Kronecker-Vec expressions we arrive at
$$\eqalign{
 M\,da &= \big((FA^T\otimes I) + (I\otimes FA)P\,\big)\,{\rm vec}(dF) \cr
 da &= M^{-1}\big((FA^T\otimes I) + (I\otimes FA)P\,\big)\,{\rm vec}(dF) \cr
\frac{\partial a}{\partial\,{\rm vec}(F)} &= M^{-1}\big((FA^T\otimes I) + (I\otimes FA)P\,\big) \cr
}$$
which, aside from my use of Latin letters, is the quantity that you asked about.
