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Kronecker product

by , PhD

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.

Table of Contents

Definition

We start with a definition.

Definition Let A be a $K	imes L$ matrix and $B$ an $M	imes N$ matrix. Then, the Kronecker product between A and $B$ is the [eq1] block matrix[eq2]where $A_{kl}$ denotes the $left( k,l
ight) $-th entry of A.

In other words, the Kronecker product $Aotimes B$ is a block matrix whose $left( k,l
ight) $-th block is equal to the $left( k,l
ight) $-th entry of A multiplied by the matrix $B$.

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

Examples

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

Example Define [eq3]and[eq4]Then, [eq5]

Example Define [eq6]and[eq7]Then, [eq8]

Example Consider the two row vectors[eq9]and[eq10]Their Kronecker product is [eq11]

Example Consider a row vector [eq12]and a column vector[eq13]Then, we have [eq14]

Example Let I be the $2	imes 2$ identity matrix and $B$ any matrix. Then, their Kronecker product is the block matrix[eq15]

Example Let a be a scalar and $B$ any matrix. Then, computing their Kronecker product is the same as multiplying $B$ by the scalar:[eq16]

Example Let A be any matrix and $b$ a scalar. Then,[eq17]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[eq18]

It suffices to provide a single counterexample.

Example Define the matrices [eq19]and [eq20]Then, we have [eq21]and[eq22]

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 [eq23]and [eq24]

Compute $Aotimes B$.

Solution

We have[eq25]

Exercise 2

Let [eq26]and [eq27]

Compute $Aotimes B$.

Solution

We have[eq28]

How to cite

Please cite as:

Taboga, Marco (2017). "Kronecker product", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/Kronecker-product.

The book

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