# Abstract Algebra/Quaternions

The set of **Quaternions** is an algebraic structure first studied by the Irish mathematician William Rowan Hamilton, in the hopes of constructing a generalization for complex numbers. When first discovered, quaternions generated a lot of excitement among mathematicians and physicists alike, for it was hoped that quaternions would provide a "unified theory" of mechanics and electromagnetism. Although these hopes proved to be unfounded, quaternions are still considered interesting as well as useful mathematical entities.

## Definition[edit]

A **Quaternion** is an ordered 4-tuple , where . A quaternion is often denoted as (Observe the analogy with complex numbers). The set of all quaternions is denoted by .

It is straightforward to define component-wise addition and scalar multiplication on , making it a real vector space.

The rule for multiplication was a product of Hamilton's ingenuity. He discovered what are known as the **Bridge-stone Equations**:

From the above equations alone, it is possible to derive rules for the pairwise multiplication of , , and :

(positive cyclic permutations)

(negative cyclic permutations).

Using these, it is easy to define a general rule for multiplication of quaternions. Because quaternion multiplication is not commutative, is not a field. However, every nonzero quaternion has a multiplicative inverse (see below), so the quaternions are an example of a non-commutative division ring. It is important to note that the non-commutative nature of quaternion multiplication makes it impossible to define the quotient of two quaternions and unambiguously, as the quantities and are generally different.

Like the more familiar complex numbers, the quaternions have a conjugation, often denoted by a superscript star: . The conjugate of the quaternion is . As is the case for the complex numbers, the product is always a positive real number equal to the sum of the squares of the quaternion's components. Using this fact, it is fairly easy to show that the multiplicative inverse of a general quaternion is given by

where division is defined since is a scalar. Note that, unlike in the complex case, the conjugate of a quaternion can be written as a polynomial in :

.

The quaternions are isomorphic to the Clifford algebra *C*ℓ_{2}(**R**) and the even subalgebra of *C*ℓ_{3}(**R**).

## Pauli Spin Matrices[edit]

Quaternions are closely related to the Pauli spin matrices of Quantum Mechanics. The Pauli matrices are often denoted as

, ,

(Where is the well known quantity of complex numbers)

The 2×2 identity matrix is sometimes taken as . It can be shown that , the real linear span of the matrices , , and , is isomorphic to the set of all quaternions, . For example, take the matrix product below:

Or, equivalently:

All three of these matrices square to the negative of the identity matrix. If we take , , , and , it is easy to see that the span of the these four matrices is "the same as" (that is, isomorphic to) the set of quaternions .

## Exercise[edit]

- Using the Bridge-stone equations, explicitly state the rule of multiplication for general quaternions, that is, given and , give the components of their product

## References[edit]

- E.T. Bell,
*Men of Mathematics*, Simon & Schuster, Inc. - The Wikipedia article on Pauli Spin Matrices