How can I decompose this matrix into the linear or bilinear product of two matrices?

Question

Given the $(m_1 + m_2) \times (n_1 + n_2)$ block matrix $$A = \begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22}\end{bmatrix}$$ where $A_{11} \in \mathbb R^{m_1 \times n_1},A_{12} \in \mathbb R^{m_1 \times n_2},A_{21} \in \mathbb R^{m_2 \times n_1},$ and $A_{22} \in \mathbb R^{m_2 \times n_2}$, and given the non-block $2 \times 2$ matrix $$B = \begin{bmatrix}b_{11} & b_{12} \\ b_{21} & b_{22}\end{bmatrix}$$ is it possible to decompose the following $(m_1 + m_2) \times (n_1 + n_2)$ block matrix $$C = \begin{bmatrix}b_{11}A_{11} & b_{12}A_{12} \\ b_{21}A_{21} & b_{22}A_{22}\end{bmatrix}$$ into some product (such as Hadamard, Kronecker, or other linear/bilinear product) of the matrices $A$ and $B$?

Answer 1

This could be interpreted as an instance of the Khatri-Rao product. In general, for any two matrices $\mathbf A$ and $\mathbf B$ partitioned into $2 \times 2$ block matrices, their Khatri-Rao product is given by $$ \mathbf{A} \ast \mathbf{B} = \left[ \begin{array} {c | c} \mathbf{A}_{11} \otimes \mathbf{B}_{11} & \mathbf{A}_{12} \otimes \mathbf{B}_{12} \\ \hline \mathbf{A}_{21} \otimes \mathbf{B}_{21} & \mathbf{A}_{22} \otimes \mathbf{B}_{22} \end{array} \right] . $$