# Abstract Algebra/Quaternions

## Contents

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

A Quaternion is an ordered 4-tuple ${\displaystyle q=(a,b,c,d)}$, where ${\displaystyle a,b,c,d\in \mathbb {R} }$. A quaternion is often denoted as ${\displaystyle q=a+bi+cj+dk}$ (Observe the analogy with complex numbers). The set of all quaternions is denoted by ${\displaystyle \mathbb {H} }$.

It is straightforward to define component-wise addition and scalar multiplication on ${\displaystyle \mathbb {H} }$, 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:

${\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1}$

From the above equations alone, it is possible to derive rules for the pairwise multiplication of ${\displaystyle i}$, ${\displaystyle j}$, and ${\displaystyle k}$:

${\displaystyle ij=k,jk=i,ki=j}$ (positive cyclic permutations)

${\displaystyle ji=-k,kj=-i,ik=-j}$ (negative cyclic permutations).

Using these, it is easy to define a general rule for multiplication of quaternions. Because quaternion multiplication is not commutative, ${\displaystyle \mathbb {H} }$ 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 ${\displaystyle p/q}$ of two quaternions ${\displaystyle p}$ and ${\displaystyle q}$ unambiguously, as the quantities ${\displaystyle pq^{-1}}$ and ${\displaystyle q^{-1}p}$ are generally different.

Like the more familiar complex numbers, the quaternions have a conjugation, often denoted by a superscript star: ${\displaystyle q^{*}}$. The conjugate of the quaternion ${\displaystyle q=a+bi+cj+dk}$ is ${\displaystyle q^{*}=a-bi-cj-dk}$. As is the case for the complex numbers, the product ${\displaystyle qq^{*}}$ 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 ${\displaystyle q}$ is given by

${\displaystyle q^{-1}={\frac {q^{*}}{qq^{*}}}}$

where division is defined since ${\displaystyle qq^{*}}$ is a scalar. Note that, unlike in the complex case, the conjugate ${\displaystyle q^{*}}$ of a quaternion ${\displaystyle q}$ can be written as a polynomial in ${\displaystyle q}$:

${\displaystyle q^{*}=-{\frac {1}{2}}(q+iqi+jqj+kqk)}$.

The quaternions are isomorphic to the Clifford algebra C2(R) and the even subalgebra of C3(R).

## Pauli Spin Matrices

Quaternions are closely related to the Pauli spin matrices of Quantum Mechanics. The Pauli matrices are often denoted as
${\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$ , ${\displaystyle \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}}$ , ${\displaystyle \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$

(Where ${\displaystyle i}$ is the well known quantity ${\displaystyle {\sqrt {-1}}}$ of complex numbers)

The 2×2 identity matrix is sometimes taken as ${\displaystyle \sigma _{0}}$. It can be shown that ${\displaystyle S}$, the real linear span of the matrices ${\displaystyle \sigma _{0}}$, ${\displaystyle i\sigma _{1}}$, ${\displaystyle i\sigma _{2}}$ and ${\displaystyle i\sigma _{3}}$, is isomorphic to the set of all quaternions, ${\displaystyle \mathbb {H} }$. For example, take the matrix product below:

${\displaystyle {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}={\begin{pmatrix}0&i\\i&0\end{pmatrix}}}$

Or, equivalently:

${\displaystyle i\sigma _{3}i\sigma _{2}=i\sigma _{1}}$

All three of these matrices square to the negative of the identity matrix. If we take ${\displaystyle 1=\sigma _{0}}$, ${\displaystyle i=i\sigma _{3}}$, ${\displaystyle j=i\sigma _{2}}$, and ${\displaystyle k=i\sigma _{1}}$, 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 ${\displaystyle \mathbb {H} }$.

## Exercise

1. Using the Bridge-stone equations, explicitly state the rule of multiplication for general quaternions, that is, given ${\displaystyle q_{1}=a_{1}+b_{1}i+c_{1}j+d_{1}k}$ and ${\displaystyle q_{2}=a_{2}+b_{2}i+c_{2}j+d_{2}k}$, give the components of their product ${\displaystyle q=q_{1}q_{2}}$

## Reference

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