The * Kronecker product*, denoted by , is a multiplication method for generating a new vector space from existing vector spaces, and therefore, new vectors from existing vectors.

Consider 2 vectors spaces, e.g. and . For in and in , we can define a new vector space, , which consists of the vector , where:

If the basis vectors for and are and respectively, the basis for is:

###### Question

Why is a new vector space?

###### Answer

An -dimensional vector space is spanned by linearly independent basis vectors. The basis vectors for and are

and consequently, the basis vectors for are

These 6 linearly independent basis vectors therefore span a 6-dimensional space.

This implies that is dimensional if is -dimensional and is -dimensional. Since is a vector space, the vectors ** **must follow the rules of addition and multiplication of a vector space. Each vector in the new vector space can then be written as a linear combination of the basis vectors , i.e. .

In general, if

then

Since the pair in is distinct for each vector, the Kronecker product results in basis vectors, which span an vector space.

As mentioned in an earlier article, a vector space is a set of objects that follows certain rules of addition and multiplication. If the objects are matrices, we have a vector space of matrices. For example, the vector spaces of matrices and generates a new vector space of matrices , where

Similarly, if the objects are functions, we have a *vector space of functions* and the Kronecker product of two vector spaces of functions and generates a new vector space of functions . If and are spanned by basis functions and basis functions respectively, is spanned by basis functions.

A vector space that is generated from two separate vector spaces has applications in quantum composite systems and in group theory.

###### Question

What is the relation between the matrix entries of *A*, *B* and *C* in ?

###### Answer

Let the matrix entries of *A*, *B* and *C* be , and respectively, where

Using the ordering convention called dictionary order, where is determined by and , and is determined by and , such that and are given by

For example, if and ,

We can then express the matrix entries of as .