\begin{definition}[Definition:Kronecker Product]
Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{p q}$ be matrices.
The '''Kronecker product''' of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \otimes \mathbf B$ and is defined as the block matrix:
:$\mathbf A \otimes \mathbf B = \begin{bmatrix}
a_{11} \mathbf B & a_{12} \mathbf B & \cdots & a_{1n} \mathbf B \\
a_{21} \mathbf B & a_{22} \mathbf B & \cdots & a_{2n} \mathbf B \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} \mathbf B & a_{m2} \mathbf B & \cdots & a_{mn} \mathbf B
\end{bmatrix}$
Writing this out in full:
:$\mathbf A \otimes \mathbf B = \begin{bmatrix}
a_{11} b_{11} & a_{11} b_{12} & \cdots & a_{11} b_{1q} & \cdots & \cdots & a_{1n} b_{11} & a_{1n} b_{12} & \cdots & a_{1n} b_{1q} \\
a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{11} b_{2q} & \cdots & \cdots & a_{1n} b_{21} & a_{1n} b_{22} & \cdots & a_{1n} b_{2q} \\
\vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\
a_{11} b_{p1} & a_{11} b_{p2} & \cdots & a_{11} b_{pq} & \cdots & \cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & \cdots & a_{1n} b_{pq} \\
\vdots & \vdots & & \vdots & \ddots & & \vdots & \vdots & & \vdots \\
\vdots & \vdots & & \vdots & & \ddots & \vdots & \vdots & & \vdots \\
a_{m1} b_{11} & a_{m1} b_{12} & \cdots & a_{m1} b_{1q} & \cdots & \cdots & a_{mn} b_{11} & a_{mn} b_{12} & \cdots & a_{mn} b_{1q} \\
a_{m1} b_{21} & a_{m1} b_{22} & \cdots & a_{m1} b_{2q} & \cdots & \cdots & a_{mn} b_{21} & a_{mn} b_{22} & \cdots & a_{mn} b_{2q} \\
\vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\
a_{m1} b_{p1} & a_{m1} b_{p2} & \cdots & a_{m1} b_{pq} & \cdots & \cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & \cdots & a_{mn} b_{pq} 
\end{bmatrix}$
Thus, if:
:$\mathbf A$ is a matrix with order $m \times n$
:$\mathbf B$ is a matrix with order $p \times q$
then $\mathbf A \otimes \mathbf B$ is a matrix with order $m p \times n q$.
\end{definition}