Definition:Kronecker Product

Let $$\mathbf{A} = \left[{a}\right]_{m n}$$ and $$\mathbf{B} = \left[{b}\right]_{p q}$$ be matrices.

The Kronecker product, or matrix direct product, of $$\mathbf{A}$$ and $$\mathbf{B}$$ is denoted $$\mathbf{A} \otimes \mathbf{B}$$ and is defined as:


 * $$\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 dimensions $$m \times n$$
 * $$\mathbf{B}$$ is a matrix with dimensions $$p \times q$$

then $$\mathbf{A} \otimes \mathbf{B}$$ is a matrix with dimensions $$m p \times n q$$.