Kronecker Product — Definition, Formula & Examples

The Kronecker product is an operation on two matrices that produces a larger block matrix by multiplying every element of the first matrix by the entire second matrix. If the first matrix is

m \times n

and the second is

p \times q

, the result is an

mp \times nq

matrix.

Given matrices $A \in \mathbb{R}^{m \times n}$ and $B \in \mathbb{R}^{p \times q}$ , the Kronecker product $A \otimes B$ is the $mp \times nq$ block matrix whose $(i,j)$ -th block is the $p \times q$ matrix $a_{ij}B$ , where $a_{ij}$ is the $(i,j)$ -th entry of $A$ .

Key Formula

A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B & \cdots & a_{1n}B \\ a_{21}B & a_{22}B & \cdots & a_{2n}B \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}B & a_{m2}B & \cdots & a_{mn}B \end{pmatrix}

Where:

$A$ = An $m \times n$ matrix
$B$ = A $p \times q$ matrix
$a_{ij}$ = The entry in row $i$, column $j$ of $A$
$\otimes$ = The Kronecker product operator

How It Works

To compute

A \otimes B

, replace each entry

a_{ij}

A

with the scaled matrix

a_{ij}B

. Arrange these blocks in the same row-column layout as

A

. The result is a single large matrix formed by concatenating all the blocks. Note that the Kronecker product is not commutative: in general

A \otimes B \neq B \otimes A

Worked Example

Problem: Compute the Kronecker product

A \otimes B

where

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

and

B = \begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix}

Step 1: Multiply each entry of

A

by the entire matrix

B

to get four

2 \times 2

blocks.

1 \cdot B = \begin{pmatrix} 0 & 5 \\ 6 & 7 \end{pmatrix}, \quad 2 \cdot B = \begin{pmatrix} 0 & 10 \\ 12 & 14 \end{pmatrix}, \quad 3 \cdot B = \begin{pmatrix} 0 & 15 \\ 18 & 21 \end{pmatrix}, \quad 4 \cdot B = \begin{pmatrix} 0 & 20 \\ 24 & 28 \end{pmatrix}

Step 2: Arrange the blocks in the same layout as

A

: block

(1,1)=1B

, block

(1,2)=2B

, block

(2,1)=3B

, block

(2,2)=4B

A \otimes B = \begin{pmatrix} 0 & 5 & 0 & 10 \\ 6 & 7 & 12 & 14 \\ 0 & 15 & 0 & 20 \\ 18 & 21 & 24 & 28 \end{pmatrix}

Answer:

A \otimes B = \begin{pmatrix} 0 & 5 & 0 & 10 \\ 6 & 7 & 12 & 14 \\ 0 & 15 & 0 & 20 \\ 18 & 21 & 24 & 28 \end{pmatrix}

Why It Matters

The Kronecker product appears frequently in quantum computing (tensor products of qubit states), signal processing, and multivariate statistics. It provides a compact way to represent operations on systems composed of independent subsystems, such as constructing covariance matrices for separable processes.

Common Mistakes

Mistake: Assuming the Kronecker product is commutative, i.e., that

A \otimes B = B \otimes A

Correction: The Kronecker product is generally not commutative. Swapping the order changes the block structure and produces a different (though permutationally similar) matrix.