TITLE: Why use the Kronecker product?
QUESTION [9 upvotes]: I have found many references on Kronecker product but I did not see any reference talking about why this way of multiplication exist and whats the intuitive use of this particular product.
Appreciate your suggestions!

REPLY [13 votes]: It is often very useful when one is solving or optimizing a function where the unknown is a matrix. This is because of the following relationship between the Kronecker product $\otimes$ and the vectorization operator $\operatorname{vec}(\cdot)$ that takes a matrix and unwinds it into a long vector:
$$\operatorname{vec}(\underbrace{AXB}_{\text{matrices}}) = \underbrace{(B^T \otimes A)}_{\text{matrix}}\underbrace{\operatorname{vec}(X)}_\text{vector}$$
For example, if you want to solve the matrix equation
$$AXB + X = C,$$
you can convert it to the following linear system:
$$(B^T \otimes A + I)\operatorname{vec}(X) = \operatorname{vec}(C).$$
More generally, matrices have both multiplicative (operator) and additive (vector space) structures, and combination of Kronecker products and vectorization provide the algebraic framework for converting back and forth between these contexts.