# Derivative of matrix using Kronecker Product

Suppose $$A(p)$$ and $$B(p)$$ are functions which map $$\mathbb{R}^{n\times m}$$ to $$\mathbb{R}^{n\times m}$$ and $$F(p)=S(I_{q}\otimes A(p))(I_{q}\otimes B(p))M$$ where $$S,M$$ are constant matrices with dimension $$qn\times qn$$ and $$qm\times qm$$ respectively. How to calculate $$\frac{d\text{vec}(F(p))}{d\text{vec}(p))}$$

Thank you very much!

Let $$\,C=AB\,$$ and note that $$(I\otimes A)(I\otimes B)= I\otimes C$$ Let's also use the convention where an uppercase letter denotes a matrix and a lowercase letter a vector, which are related by vectorization, e.g. $$a={\rm vec}(A),\,\,\,b={\rm vec}(B),\,\,\,c={\rm vec}(C),\,\,\,etc$$ Finally, we'll need the SVD of the rightmost matrix in your function. $$M = \sum_k \sigma_ku_kv_k^T$$ Break the function into components (corresponding to an SVD component). \eqalign{ F_k &= S(I\otimes C)u_kv_k^T\sigma_k \cr &= S\,{\rm vec}(CU_k)\,v_k^T\sigma_k \cr &= S(U_k^T\otimes I)\,cv_k^T\sigma_k \cr f_k &= \Big(\sigma_kv_k\otimes\big(S(U_k^T\otimes I)\big)\Big)\,c \cr } Now find the derivative of this (vectorized) component matrix. \eqalign{ \frac{\partial f_k}{\partial p} &= \Big(\sigma_kv_k\otimes\big(S(U_k^T\otimes I)\big)\Big)\frac{\partial c}{\partial p} \cr } And since $$F = \sum_k F_k\,$$ we have our answer \eqalign{ \frac{\partial f}{\partial p} &= \sum_k\Big(\sigma_kv_k\otimes\big(S(U_k^T\otimes I)\big)\Big)\frac{\partial c}{\partial p} } I'll leave it to you to work out the derivative on the far RHS, seeing as you told us nothing about the functional form of $$A(P)$$ and $$B(P)$$ and therefore of $$C(P)$$.