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.

We start with a definition.

Definition Let be a matrix and an matrix. Then, the Kronecker product between and is the block matrixwhere 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 .

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

Example Define andThen,

Example Define andThen,

Example Consider the two row vectorsandTheir Kronecker product is

Example Consider a row vector and a column vectorThen, 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, that is, in general

It suffices to provide a single counterexample.

Example Define the matrices and Then, we have and

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

Below you can find some exercises with explained solutions.

Let and

Compute .

Solution

We have

Let and

Compute .

Solution

We have

Please cite as:

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

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