Kronecker product

Definition

Let $F$ be a field, let $A = (a_{ij})$ be an $m \times n$ matrix over $F$, and let $B$ be a $p \times q$ matrix over $F$. The Kronecker product $A \otimes B$ is the $mp \times nq$ block matrix $$A \otimes B = \begin{pmatrix} a_{11} B & \cdots & a_{1n} B \\ \vdots & \ddots & \vdots \\ a_{m1} B & \cdots & a_{mn} B \end{pmatrix}.$$

Remarks

It satisfies $(A \otimes B)(C \otimes D) = (AC) \otimes (BD)$ when the products $AC$ and $BD$ are defined. If $A$ and $B$ are invertible, then $(A \otimes B)^{-1} = A^{-1} \otimes B^{-1}$. It is not commutative in general, but $A \otimes B$ and $B \otimes A$ are permutation equivalent. It is the matrix of the tensor product of two linear maps, in the tensor product bases ordered suitably.

Explanations

L1 human

The Kronecker product replaces each entry of $A$ with that entry times the whole matrix $B$. For example, $I_2 \otimes B$ is the block diagonal matrix with two copies of $B$. It is the matrix form of the tensor product.