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Commutativity of quaternions
On the non-commutativity of the quaternion algebra; with examples

Prerequisites

To understand this, you should read about the Cayley-Dickson construction

Introduction

To show that the quaternion algebra isn’t commutative, all that is needed is one example of two quaternions (a, b) and (c, d) such that

(a, b) (c, d) is not equal to (c, d) (a, b).

Another question would be, for any right multiplication by a quaternion, is there an equivalent left multiplication? That is, given a quaternion r, is there a (perhaps different) quaternion s such that

r q = q s

for all quaternions q?

The answer to this question is “no”. That is, there are two distinct (but overlapping) sets of linear transformations of the quaternions, corresponding to multiplication by a fixed quaternion on the left and on the right.

Just which quaternions commute with all other quaternions? Below, you will find the answer is: precisely the real quaternions.

Example: non-commutative quaternions

Some quaternions do commute.

Any quaternion corresponding to a real number commutes with every other quaternion. The quaternions corresponding to complex numbers will commute with one another, because the complex numbers are commutative. An example of quaternions that don’t commute will therefore consist of two quaternions that don’t both correspond to complex numbers.

With that in mind, it’s easy to find an example of quaternions that don’t commute. Using the formula for multiplication of quaternions,

(0, i) (i, 0) = (0, −1),

but

(i, 0) (0, i) = (0, 1).

Notice that this also shows that quaternions corresponding to complex numbers don’t commute with all other quaternions. (It’s not hard to show that the complexes commute only with one another).

In algebraic terminology, the reals are said to be the center of the group of quaternions under multiplication. The complex numbers are an abelian subgroup of the quaternions under multiplication.

Example: left quaternion multiplication with no right equivalent

There is no quaternion which has the same effect when multiplied on the right of all complex numbers, as does multiplication on the left by the pure imaginary number i.

It is interesting to compare multiplication on the left and right by i:

(i, 0) (a, b) = (i a, b i) ;
(a, b) (i, 0) = (a i, b i*)

This says that multiplication on the left by i effects a rotation of the two complex components of a quaternion in the same direction by 90°, while multiplication on the right by i effects rotation of the two complex components in opposite directions by 90°.

Of course, the very possibility of doing two independent rotations at once first appears in 4. It is a very hard thing to visualize, but something very characteristic of geometry in that space.

Only real quaternions have right equivalents

It is a matter of calculation to show that left multiplication by a quaternion has a right equivalent if and only if it is real, and that then it is its own right equivalent.

One approach that doesn’t require too much calculation is to write out the equation

a = r a s−1

using the representation of quaternions as a pair of complex numbers, apply multiplication formula, use the commutativity of complex numbers, and finally, the fact that if

z = p z + q z*

for all complex z, then p = 1 and q = 0.

Multiplication on the left and right by quaternions commute with one another

Although multiplication among quaternions is not generally commutative, the operation of multiplication on the left commutes with that on the right.

In the usual notation of quaternion algebra, this statement is the same as that of associativity of quaternion multiplication: for any quaternion x,

(q x) p = q (x p).

That is, in the product q x p, it is immaterial whether the multiplication on the left is performed first, or the multiplication on the right.

It is often convenient to employ other notations for quaternion multiplication, in which this property is not so obvious. For example, if QL represents the linear operator of multiplication on the left by quaternion q, and PR represents the linear operator of multiplication on the right by quaternion p, the above identity reads

QL PR x = PR QL x.

In algebraic terms, the operations of quaternion multiplication on the left and right are said to comprise two quasinormal subgroups of the group of multiplications by quaternions: they each commute with all other proper subgroups of operations of multiplication by quaternions, although they do not commute with the whole group, as they do not commute with themselves.

See also

The Geometry of Quaternions: Geometry of non-commutativity