What is the operator that can combine two 2x2 matrices into one 4x4 matrix as shown below?
A = $\begin{bmatrix}a&b\\c&d\end{bmatrix}$
X = $\begin{bmatrix}x&y\\z&w\end{bmatrix}$
A ? X = $\begin{bmatrix}aX&bX\\cX&dX\end{bmatrix} = \begin{bmatrix}ax&ay&bx&by\\az&aw&bz&bw\\cx&cy&dx&dy\\cz&cw&dz&dw\end{bmatrix}$
This is called the Kronecker product. It's one manifestation of the tensor product from abstract algebra, and it's usually denoted by $\otimes$.