Solve matrix equations involving vectorization and Kronecker product

Question

I want to find solutions for matrices $A\in \mathbb{R}^{m\times n}$ and $B\in \mathbb{R}^{n\times m}$ in the following equations:

$$ \left\{ \begin{matrix} A^TM_1 = A^T(AB\odot M_2) \\ M_1B^T = (AB\odot M_2) B^T \end{matrix} \right., $$ where $M_1,M_2\in \mathbb{R}^{m\times m}$ are known invertible matrices, and $\odot$ represents Hardamard product. Through vectorization we can have the following equations: $$ \left\{ \begin{matrix} (I\otimes A^T) \operatorname{vec}(M_1) = (I \otimes A^T)\operatorname{diag}(\operatorname{vec}(M_2)) (I \otimes A)\operatorname{vec}(B) \\ (B\otimes I) \operatorname{vec}(M_1) = (B\otimes I)\operatorname{diag}(\operatorname{vec}(M_2)) (B^T\otimes I)\operatorname{vec}(A) \end{matrix} \right., $$ where $\otimes$ represents Kronecker product, $\operatorname{diag}()$ denotes the diagonal matrix. However, I'm stuck on this step. We get $\operatorname{vec}(B)$ in the first equation, but how to transform it into $(B\otimes I)$?

Answer 1

To expand on my comment, reduce the two equations to a single equation $$AB \;=\; C \;=\; (M_1\oslash M_2)$$ To solve this equation, randomly initialize $B\:$ then iterate using alternating pseudoinverses $$\eqalign{ &B \;\;= &B_0 \\ &k \;\;= &0 \\ &{\rm repeat} \\ &&A_{k+1} = CB_k^+ \\ &&B_{k+1} = A_{k+1}^+C \\ &&k = k+{\tt1} \\ &{\rm until}&\|C-A_kB_k\| \le 10^{-12} \\ }$$ If $m\le n$ this will converge in a single iteration (assuming that a solution exists).
Each $B_0$ usually leads to a different solution.
Since $AB=(AZ)(Z^{-1}B)$ for any invertible $Z$, an infinite number of solutions are possible.