 StatLect

# Kronecker product

The Kronecker product is an operation that transforms two matrices into a larger matrix that contains all the possible products of the entries of the two matrices. It possesses several properties that are often used to solve difficult problems in linear algebra and its applications. ## Definition

Definition Let be a matrix and an matrix. Then, the Kronecker product between and is the block matrix where denotes the -th entry of .

In other words, the Kronecker product is a block matrix whose -th block is equal to the -th entry of multiplied by the matrix .

Note that, unlike the ordinary product between two matrices, the Kronecker product is defined regardless of the dimensions of the two matrices and .

## Examples

Although the concept is relatively simple, it is often beneficial to see several examples of Kronecker products.

Example Define and Then, Example Define and Then, Example Consider the two row vectors and Their Kronecker product is Example Consider a row vector and a column vector Then, we have Example Let be the identity matrix and any matrix. Then, their Kronecker product is the block matrix Example Let be a scalar and any matrix. Then, computing their Kronecker product is the same as multiplying by the scalar: Example Let be any matrix and a scalar. Then, where we have used the definition of multiplication of a matrix by a scalar.

## The Kronecker product is not commutative

The Kronecker product is not commutative, that is, in general It suffices to provide a single counterexample.

Example Define the matrices and Then, we have and ## Properties

The many properties of Kronecker products will be discussed in the lecture on the Properties of the Kronecker product.

## Solved exercises

Below you can find some exercises with explained solutions.

### Exercise 1

Let and Compute .

Solution

We have ### Exercise 2

Let and Compute .

Solution

We have 