# 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)

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(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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle p/q}**
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 :

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.

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

, , **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \sigma_3=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}}**

(Where is the well known quantity of complex numbers)

The 2×2 identity matrix is sometimes taken as . It can be shown that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): S**
, the real linear span of the matrices , , and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle i\sigma_3}**
, is isomorphic to the set of all quaternions, . For example, take the matrix product below:

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Or, equivalently:

All three of these matrices square to the negative of the identity matrix. If we take **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle 1=\sigma_0}**
, , , 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

## Reference[edit]

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