Recognizing a $4\times4$ matrix

Question

Suppose I have $$ \Sigma=\begin{pmatrix}\sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22}\end{pmatrix},\quad S=\begin{pmatrix}s_{11} & s_{12} \\ s_{21} & s_{22}\end{pmatrix} $$ both of which are actually covariance matrices of two $2\times 1$ random vectors. So, in particular, $\Sigma$ and $S$ are symmetric. Now I have a $4\times 4$ matrix: $$ A=\begin{pmatrix} \sigma_{11}s_{11} & \sigma_{11}s_{12} & \sigma_{12}s_{11} & \sigma_{12}s_{12} \\ \sigma_{12}s_{11} & \sigma_{12}s_{12} & \sigma_{22}s_{11} & \sigma_{22}s_{12} \\ \sigma_{11}s_{12} & \sigma_{11}s_{22} & \sigma_{12}s_{12} & \sigma_{12}s_{22} \\ \sigma_{12}s_{12} & \sigma_{12}s_{22} & \sigma_{22}s_{12} & \sigma_{22}s_{22} \end{pmatrix} $$ I would like to write $A$ as a function of $\Sigma$ and $S$ (without referring to $\sigma$'s and $s$') using matrix operations ($\otimes$, multiplication, inversion, addition, and maybe something else I cannot think of). How do I do that please?

Answer 1

You can write your matrix as $A = (S \otimes \Sigma)P$, where $\otimes$ denotes the Kronecker product and $$ P = \pmatrix{1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1} $$

Answer 2

I might be wrong, but one way to do it may be through combining Kronecker product and Hadamard product:

$$ \biggl(\begin {bmatrix} 1 & 1 \end {bmatrix} \otimes S \otimes \begin{bmatrix} 1 \\ 1 \end {bmatrix}\biggr) \circ \biggl(\begin{bmatrix} 1 \\ 1 \end {bmatrix}\ \otimes \Sigma \otimes \begin {bmatrix} 1 & 1 \end {bmatrix}\biggr) $$