quaternion

Definition

There are two ways to define the quaternions, which are often denoted with the symbol .

Over the real numbers

The quaternions are a 4-dimensional vector space over the real numbers with the following “standard basis” as matrices.

These elements follow the multiplication rules (which can be seen using matrix multiplication on the basis defined above):

In this way, every quaternion can be written in the form

We call the set

the pure quaternions. So thus, the set of quaternions can be written as

and in this form, conjugation works similarly to in the complex numbers.

Over the complex numbers

As a complex vector space, the quaternions are 2-dimensional. They can be written in matrices of the form

In this way, the complex numbers embed into

In this way, we can talk about the matrices as having the “standard basis”

Using this basis, the multiplication rules are

It is obvious this works since if we multiply the matrices we have

Properties

Multiplication is obviously not commutative in the quaternions (since ) so it is not a field (which makes sense since the only field extension of is ). However, the quaternions do form a division algebra over .