# Definition:Kronecker Product

## Definition

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 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 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 $m p \times n q$.

## Source of Name

This entry was named for Leopold Kronecker.